m4

.gitignore

@@ -0,0 +1,5 @@
+bin/
+obj/
+/packages/
+riderModule.iml
+/_ReSharper.Caches/

VirtualChemistry.csproj

@@ -0,0 +1,13 @@
+<Project Sdk="Microsoft.NET.Sdk">
+
+    <PropertyGroup>
+        <TargetFramework>net6.0</TargetFramework>
+        <ImplicitUsings>enable</ImplicitUsings>
+        <Nullable>enable</Nullable>
+    </PropertyGroup>
+
+    <ItemGroup>
+      <ProjectReference Include="..\RiderProjects\Skeleton\Skeleton.csproj" />
+    </ItemGroup>
+
+</Project>

VirtualChemistry.sln

@@ -0,0 +1,27 @@
+
+Microsoft Visual Studio Solution File, Format Version 12.00
+Project("{FAE04EC0-301F-11D3-BF4B-00C04F79EFBC}") = "VirtualChemistry", "VirtualChemistry.csproj", "{36979A00-F434-4001-A80E-ECBF92A0AB5E}"
+EndProject
+Project("{FAE04EC0-301F-11D3-BF4B-00C04F79EFBC}") = "Skeleton", "..\RiderProjects\Skeleton\Skeleton.csproj", "{CB0ED85F-CAC0-4B53-96A8-16048A928890}"
+EndProject
+EndProject
+Global
+	GlobalSection(SolutionConfigurationPlatforms) = preSolution
+		Debug|Any CPU = Debug|Any CPU
+		Release|Any CPU = Release|Any CPU
+	EndGlobalSection
+	GlobalSection(ProjectConfigurationPlatforms) = postSolution
+		{36979A00-F434-4001-A80E-ECBF92A0AB5E}.Debug|Any CPU.ActiveCfg = Debug|Any CPU
+		{36979A00-F434-4001-A80E-ECBF92A0AB5E}.Debug|Any CPU.Build.0 = Debug|Any CPU
+		{36979A00-F434-4001-A80E-ECBF92A0AB5E}.Release|Any CPU.ActiveCfg = Release|Any CPU
+		{36979A00-F434-4001-A80E-ECBF92A0AB5E}.Release|Any CPU.Build.0 = Release|Any CPU
+		{917762E5-D59D-4724-8F1A-73F739573E99}.Debug|Any CPU.ActiveCfg = Debug|Any CPU
+		{917762E5-D59D-4724-8F1A-73F739573E99}.Debug|Any CPU.Build.0 = Debug|Any CPU
+		{917762E5-D59D-4724-8F1A-73F739573E99}.Release|Any CPU.ActiveCfg = Release|Any CPU
+		{917762E5-D59D-4724-8F1A-73F739573E99}.Release|Any CPU.Build.0 = Release|Any CPU
+		{CB0ED85F-CAC0-4B53-96A8-16048A928890}.Debug|Any CPU.ActiveCfg = Debug|Any CPU
+		{CB0ED85F-CAC0-4B53-96A8-16048A928890}.Debug|Any CPU.Build.0 = Debug|Any CPU
+		{CB0ED85F-CAC0-4B53-96A8-16048A928890}.Release|Any CPU.ActiveCfg = Release|Any CPU
+		{CB0ED85F-CAC0-4B53-96A8-16048A928890}.Release|Any CPU.Build.0 = Release|Any CPU
+	EndGlobalSection
+EndGlobal

global.json

@@ -0,0 +1,7 @@
+{
+  "sdk": {
+    "version": "6.0.0",
+    "rollForward": "latestMinor",
+    "allowPrerelease": false
+  }
+}

src/Chemistry/Atoms/Implements/AAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom 12
+/// </summary>
+public class AAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.A;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 12;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateA;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateA;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateA;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassA;
+}

src/Chemistry/Atoms/Implements/BaseAtom.cs

@@ -0,0 +1,386 @@
+using System.Text;
+using Skeleton.Algebra.AdditionalProperties.Extensions;
+using Skeleton.Algebra.DimensionProviders;
+using Skeleton.Constants;
+using Skeleton.DataStructure;
+using Skeleton.Utils.Helpers;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Chemistry.Bonds.Resolver;
+using VirtualChemistry.Chemistry.Compounds.Implements;
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+using VirtualChemistry.Constants;
+using VirtualChemistry.DataStructure.Packs;
+
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// meta atom
+/// </summary>
+public abstract class BaseAtom
+{
+    /// <summary>
+    /// 
+    /// </summary>
+    protected BaseAtom()
+    {
+        LogEigenState = new(_ => EigenStateMatrix.Log());
+        BondInformationMatrix = new(
+            x => Bonds
+                .Select(bond => bond.LogState.GetFrom(x))
+                .DefaultIfEmpty(LieSU3.Zero)
+                .SpLieSum()
+                .Exp()
+        );
+
+        LogKernelMatrix = new(_ => KernelState.Log());
+        StateMatrix = new(x => BondInformationMatrix.GetFrom(x).ConjugateOn(EigenStateMatrix));
+        LogStateMatrix = new(x => BondInformationMatrix.GetFrom(x).ConjugateOn(LogEigenState.GetFrom(x)));
+        StructureMatrix = new(x =>
+            {
+                var h = Bonds
+                    .Select(b => b.LogConnectionMatrix.GetFrom(x))
+                    .DefaultIfEmpty(LieSU3.Zero)
+                    .Union(new[] { LogStateMatrix.GetFrom(x), LogKernelMatrix.GetFrom(x) });
+                var s = h.SpLieSum().Exp();
+                return s;
+            }
+        );
+        
+        EnergyDistributionFactor = new CacheItem<double>(
+            x =>
+            {
+                C3 v = StructureMatrix.GetFrom(x).Hermitian() * ChemistryConstant.AtomEnergyDistributionVector;
+                return ChemistryConstant.AtomEnergyDistributionSpectrum.Rayleigh<IDim3>(v);
+            }
+        );
+    }
+
+    /// <summary>
+    /// singleton to represent no atom
+    /// </summary>
+    public static readonly BaseAtom Null = new NullAtom();
+    /// <summary>
+    /// energy holds in the atom
+    /// </summary>
+    public double Energy
+    {
+        get => Bonds.Sum(bond => bond.Energy);
+        set
+        {
+            double energyFactor = Bonds.Sum(bond => bond.EnergyDistributionFactor.Get);
+            foreach (BaseBond bond in Bonds)
+                bond.Energy = value * bond.EnergyDistributionFactor.Get / energyFactor;
+        }
+    }
+
+    /// <summary>
+    /// describe how energy in molecular is distributed in this atom
+    /// </summary>
+
+
+    public CacheItem<double> EnergyDistributionFactor { get; }
+
+    /// <summary>
+    /// element name/ atom name
+    /// </summary>
+    public abstract string Element { get; }
+    /// <summary>
+    /// atomic number, redundant information as element
+    /// </summary>
+    public abstract int AtomicNumber { get; }
+    /// <summary>
+    /// atomic mass, wont change as long as atomic number is fixed
+    /// </summary>
+    public abstract double AtomicMass { get; }
+    /// <summary>
+    /// compound of this atom
+    /// </summary>
+    public Compound Compound { get; set; } = Compound.Null;
+    /// <summary>
+    /// homogeneous mixture of its compound
+    /// </summary>
+    public HomogeneousMixture HomogeneousMixture => Compound.HomogeneousMixture;
+    /// <summary>
+    /// bonds
+    /// </summary>
+    public HashSet<BaseBond> Bonds { get; set; } = new HashSet<BaseBond>();
+    /// <summary>
+    /// bonds that already connected with other bond
+    /// </summary>
+    public IEnumerable<BaseBond> ConnectedBonds => Bonds.Where(x => x.Connected);
+    
+
+    /// <summary>
+    /// bonds that not connected with any other bond yet
+    /// </summary>
+    public IEnumerable<BaseBond> UnconnectedBonds => Bonds.Where(x => !x.Connected);
+    /// <summary>
+    /// atoms that connected to this atom directly
+    /// </summary>
+    public IEnumerable<BaseAtom> ConnectedAtoms => ConnectedBonds
+        .Select(bond => bond.ConnectedBond.Atom);
+    /// <summary>
+    /// mass of all directly connected atoms
+    /// </summary>
+    public double GroupMass => ConnectedAtoms
+        .Select(atom => atom.AtomicMass)
+        .DefaultIfEmpty(0d)
+        .Sum();
+    /// <summary>
+    /// if there is a path to specific atom
+    /// private use only
+    /// </summary>
+    /// <param name="atom"></param>
+    /// <param name="visited"></param>
+    /// <returns></returns>
+    private bool HasPathTo(BaseAtom atom, HashSet<BaseAtom> visited)
+    {
+        if (atom == this)
+            return true;
+        if (ConnectedAtoms.Contains(atom))
+            return true;
+        visited.Add(this);
+        return ConnectedAtoms
+            .Where(connectedAtom => !visited.Contains(connectedAtom))
+            .Any(connectedAtom => connectedAtom.HasPathTo(atom, visited));
+    }
+    /// <summary>
+    /// if there is a path to a specific atom
+    /// </summary>
+    /// <param name="atom"></param>
+    /// <returns></returns>
+    public bool HasPathTo(BaseAtom atom) => HasPathTo(atom, new HashSet<BaseAtom>());
+    /// <summary>
+    /// get the compound by connections
+    /// </summary>
+    public Compound GrabCompound
+    {
+        get
+        {
+            Compound res = new ();
+            res.Atoms.Add(this);
+            int aCount = 0;
+            while (aCount != res.Atoms.Count)
+            {
+                aCount = res.Atoms.Count;
+                foreach (BaseAtom a in res.Atoms
+                             .SelectMany(a => a.ConnectedAtoms)
+                             .Where(a => !res.Atoms.Contains(a))
+                             .ToArray()
+                         )
+                    res.Atoms.Add(a);
+            }
+
+            res.HomogeneousMixture = HomogeneousMixture.Null;
+            res.Amount = 0d;
+            return res;
+        }
+    }
+    /// <summary>
+    /// how far is an atom from this one(directly connected atoms has distance 1)
+    /// </summary>
+    /// <param name="atom"></param>
+    /// <param name="visited"></param>
+    /// <returns></returns>
+    private int DistanceTo(BaseAtom atom, HashSet<BaseAtom> visited)
+    {
+        if (Compound != atom.Compound)
+            return -1;
+        if (atom == this)
+            return 0;
+        if(Compound.CachedDistance(this, atom) > 0)
+            return Compound.CachedDistance(this, atom);
+        if (ConnectedAtoms.Contains(atom))
+            return 1;
+        visited.Add(this);
+        int distance = ConnectedAtoms
+            .Where(connectedAtom => !visited.Contains(connectedAtom))
+            .Select(connectedAtom => connectedAtom.DistanceTo(atom, visited))
+            .DefaultIfEmpty(int.MaxValue)
+            .Min();
+        Compound.CacheDistance(this, atom, distance + 1);
+        return distance + 1;
+    }
+    /// <summary>
+    /// how far is an atom from this one(directly connected atoms has distance 1)
+    /// </summary>
+    /// <param name="atom"></param>
+    /// <returns></returns>
+    public int DistanceTo(BaseAtom atom) => DistanceTo(atom, new HashSet<BaseAtom>());
+
+    /// <summary>
+    /// all connections of this atom
+    /// </summary>
+    public IEnumerable<ConnectionInfo> Connections =>
+        Bonds
+            .Where(bond => bond.Connected)
+            .Select(bond => new ConnectionInfo(bond, bond.ConnectedBond, bond.ConnectedBond.Atom));
+    /// <summary>
+    /// distance info as map
+    /// </summary>
+    /// <returns></returns>
+    private Dictionary<int, HashSet<BaseAtom>> DistanceMap()
+    {
+        Dictionary<int, HashSet<BaseAtom>> res = new Dictionary<int, HashSet<BaseAtom>>();
+        foreach (BaseAtom atom in Compound.Atoms)
+        {
+            int dist = DistanceTo(atom);
+            if(!res.ContainsKey(dist))
+                res.Add(dist, new HashSet<BaseAtom>());
+            res[dist].Add(atom);
+        }
+        return res;
+    }
+
+    /// <summary>
+    /// order bonds by selectors
+    /// </summary>
+    public IOrderedEnumerable<BaseBond> OrderedBonds =>
+        Bonds
+            .OrderBy(bond => bond.Connected)
+            .ThenBy(bond => bond.BondNumber)
+            .ThenBy(bond => bond.Connected ? bond.ConnectedBond.BondNumber : 0)
+            .ThenBy(bond => bond.Connected ? bond.ConnectedBond.Atom.AtomicNumber : 0);
+    /// <summary>
+    /// kernel state matrix of the atom, wont change
+    /// </summary>
+    public abstract SU3 KernelState { get; }
+    /// <summary>
+    /// adj matrix wont change
+    /// </summary>
+    public abstract SU3 AdjointState { get; }
+    /// <summary>
+    /// eig matrix wont change
+    /// </summary>
+    public abstract SU3 EigenStateMatrix { get; }
+
+    /// <summary>
+    /// will never expire
+    /// </summary>
+    public CacheItem<LieSU3> LogKernelMatrix { get; }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public CacheItem<SU3> BondInformationMatrix { get; }
+
+    /// <summary>
+    /// will never expire
+    /// </summary>
+    public CacheItem<LieSU3> LogEigenState { get; }
+
+    /// <summary>
+    /// state matrix
+    /// </summary>
+    public CacheItem<SU3> StateMatrix { get; }
+
+    /// <summary>
+    /// log state matrix
+    /// </summary>
+    public CacheItem<LieSU3> LogStateMatrix { get; }
+
+    /// <summary>
+    /// get an unconnected bond of specific type
+    /// </summary>
+    /// <param name="bondType"></param>
+    /// <returns></returns>
+    public BaseBond NextUnconnected(string bondType) =>
+        Bonds
+            .Where(bond => bond.BondType == bondType)
+            .FirstOrDefault(bond => !bond.Connected, BaseBond.Null);
+    /// <summary>
+    /// get an unconnected bond of specific type, except for bonds in exclude set
+    /// </summary>
+    /// <param name="bondType"></param>
+    /// <param name="exclude"></param>
+    /// <returns></returns>
+    public BaseBond NextUnconnected(string bondType, HashSet<BaseBond> exclude) =>
+        Bonds
+            .Where(bond => bond.BondType == bondType)
+            .Where(bond => !exclude.Contains(bond))
+            .FirstOrDefault(bond => !bond.Connected, BaseBond.Null);
+    /// <summary>
+    /// get an unconnected bond of specific type that is not the excluded bond
+    /// </summary>
+    /// <param name="bondType"></param>
+    /// <param name="exclude"></param>
+    /// <returns></returns>
+    public BaseBond NextUnconnected(string bondType, BaseBond exclude) =>
+        Bonds
+            .Where(bond => bond.BondType == bondType)
+            .Where(bond => exclude != bond)
+            .FirstOrDefault(bond => !bond.Connected, BaseBond.Null);
+
+    /// <summary>
+    /// structure matrix
+    /// </summary>
+    public CacheItem<SU3> StructureMatrix { get; }
+
+    /// <summary>
+    /// the representation of the compound relative to a center atom
+    /// </summary>
+    /// <param name="center"></param>
+    /// <param name="visited"></param>
+    /// <returns></returns>
+    public StringBuilder IsoRepresentation(BaseAtom center, HashSet<BaseBond> visited)
+    {
+        HashSet<BaseBond> toVisit = Bonds
+            .Where(b => b.Connected)
+            .Where(b => !visited.Contains(b))
+            .Where(b => !visited.Contains(b.ConnectedBond))
+            .ToHashSet();
+        foreach (BaseBond b in toVisit)
+        {
+            visited.Add(b);
+            visited.Add(b.ConnectedBond);
+            
+        }
+
+        Dictionary<BaseBond, StringBuilder> subRepresentations = new ();
+        foreach (BaseBond b in toVisit)
+            subRepresentations[b] =
+                b.ConnectedBond.Atom.IsoRepresentation(center, new HashSet<BaseBond>(visited));
+        
+        StringBuilder res = new StringBuilder($"{Element}#{DistanceTo(center)}-[");
+        IEnumerable<BaseBond> visitOrder = toVisit
+            .GroupedOrderBy(bond => bond.BondNumber)
+            .GroupedThenBy(bond => bond.ConnectedBond.BondNumber)
+            .GroupedThenBy(bond => bond.ConnectedBond.Atom.AtomicNumber)
+            .GroupedThenBy(bond => bond.ConnectedBond.Atom.ConnectedAtoms.Count())
+            .GroupedThenBy(bond => subRepresentations[bond].ToString())
+            .SelectMany(g => g);
+        foreach (BaseBond b in visitOrder)
+        {
+            res.Append($"<{b.BondType}={b.ConnectedBond.BondType}*");
+            res.Append(subRepresentations[b]);
+            res.Append(">");
+        }
+        res.Append("]");
+        
+        return res;
+    }
+    /// <summary>
+    /// the representation of the compound with this atom as center
+    /// </summary>
+    public string FullRepresentation => IsoRepresentation(this, new HashSet<BaseBond>()).ToString();
+    /// <summary>
+    /// initializer 
+    /// </summary>
+    public void C()
+    {
+        Bonds = new HashSet<BaseBond>();
+        for (int bondNumber = 1; bondNumber <= 7; bondNumber++)
+        {
+            int xDiff = AtomicNumber - AlgebraConstant.Ludic(bondNumber);
+            if (AtomicNumber % AlgebraConstant.Ludic(bondNumber) != 0 &&
+                (xDiff < 0 || xDiff % 7 != 0))
+                continue;
+            for (int idx = 0; idx < ChemistryConstant.BondMaxNumbers(bondNumber); idx++)
+                Bonds.Add(BondResolver.Resolve(bondNumber));
+        }
+
+        foreach (BaseBond b in Bonds)
+            b.Atom = this;
+    }
+    
+}

src/Chemistry/Atoms/Implements/CxAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 13
+/// </summary>
+public class CxAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.Cx;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 13;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateCx;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateCx;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateCx;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassCx;
+}

src/Chemistry/Atoms/Implements/DAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 4
+/// </summary>
+public class DAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.D;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 4;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateD;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateD;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateD;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassD;
+}

src/Chemistry/Atoms/Implements/EAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 10
+/// </summary>
+public class EAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.E;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 10;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateE;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateE;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateE;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassE;
+}

src/Chemistry/Atoms/Implements/EsAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 3
+/// </summary>
+public class EsAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.Es;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 3;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateEs;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateEs;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateEs;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassEs;
+}

src/Chemistry/Atoms/Implements/KAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 7
+/// </summary>
+public class KAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.K;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 7;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateK;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateK;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateK;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassK;
+}

src/Chemistry/Atoms/Implements/MAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 6
+/// </summary>
+public class MAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.M;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 6;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateM;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateM;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateM;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassM;
+}

src/Chemistry/Atoms/Implements/NullAtom.cs

@@ -0,0 +1,30 @@
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # -1, the atom used to represent no atom
+/// </summary>
+public sealed class NullAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public NullAtom()
+    {
+    }
+
+    /// <inheritdoc />
+    public override string Element => "Null";
+
+    /// <inheritdoc />
+    public override int AtomicNumber => throw new Exception();
+
+    /// <inheritdoc />
+    public override double AtomicMass => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 KernelState => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => throw new Exception();
+}

src/Chemistry/Atoms/Implements/PAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 8
+/// </summary>
+public class PAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.P;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 8;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateP;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateP;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateP;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassP;
+}

src/Chemistry/Atoms/Implements/QAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 1
+/// </summary>
+public class QAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.Q;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 1;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateQ;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateQ;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateQ;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassQ;
+}

src/Chemistry/Atoms/Implements/RAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 2
+/// </summary>
+public class RAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.R;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 2;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateR;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateR;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateR;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassR;
+}

src/Chemistry/Atoms/Implements/SoAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 9
+/// </summary>
+public class SoAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.So;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 9;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateSo;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateSo;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateSo;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassSo;
+}

src/Chemistry/Atoms/Implements/UAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 11
+/// </summary>
+public class UAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.U;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 11;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateU;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateU;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateU;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassU;
+}

src/Chemistry/Atoms/Implements/UeAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 5
+/// </summary>
+public class UeAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.Ue;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 5;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateUe;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateUe;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateUe;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassUe;
+}

src/Chemistry/Atoms/Implements/VAtom.cs

@@ -0,0 +1,26 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+/// <summary>
+/// atom # 14
+/// </summary>
+public class VAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.V;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 14;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateV;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateV;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateV;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassV;
+}

src/Chemistry/Atoms/Implements/ViAtom.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Implements;
+
+/// <summary>
+/// atom # 15
+/// </summary>
+public class ViAtom : BaseAtom
+{
+    /// <inheritdoc />
+    public override string Element => ChemistryConstant.Elements.Vi;
+
+    /// <inheritdoc />
+    public override int AtomicNumber => 15;
+
+    /// <inheritdoc />
+    public override SU3 AdjointState => ChemistryConstant.AtomAdjoints.AtomAdjointStateVi;
+
+    /// <inheritdoc />
+    public override SU3 KernelState => ChemistryConstant.AtomKernels.AtomKernelStateVi;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.AtomEigenStates.AtomEigenStateVi;
+
+    /// <inheritdoc />
+    public override double AtomicMass => ChemistryConstant.AtomicMasses.AtomicMassVi;
+}

src/Chemistry/Atoms/Resolver/AtomResolver.cs

@@ -0,0 +1,72 @@
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Atoms.Resolver;
+
+/// <summary>
+/// get an atom instance from descriptions 
+/// </summary>
+public static class AtomResolver
+{
+    /// <summary>
+    /// create an atom from string
+    /// </summary>
+    /// <param name="element"></param>
+    /// <returns></returns>
+    /// <exception cref="Exception"></exception>
+    public static BaseAtom Resolve(string element)
+    {
+        BaseAtom res = element switch
+        {
+            ChemistryConstant.Elements.Q => new QAtom(),
+            ChemistryConstant.Elements.R => new RAtom(),
+            ChemistryConstant.Elements.Es => new EsAtom(),
+            ChemistryConstant.Elements.D => new DAtom(),
+            ChemistryConstant.Elements.Ue => new UeAtom(),
+            ChemistryConstant.Elements.M => new MAtom(),
+            ChemistryConstant.Elements.K => new KAtom(),
+            ChemistryConstant.Elements.P => new PAtom(),
+            ChemistryConstant.Elements.So => new SoAtom(),
+            ChemistryConstant.Elements.E => new EAtom(),
+            ChemistryConstant.Elements.U => new UAtom(),
+            ChemistryConstant.Elements.A => new AAtom(),
+            ChemistryConstant.Elements.Cx => new CxAtom(),
+            ChemistryConstant.Elements.V => new VAtom(),
+            ChemistryConstant.Elements.Vi => new ViAtom(),
+            _ => throw new Exception($"TODO-----{element}")
+        };
+        res.C();
+        return res;
+    }
+    /// <summary>
+    /// create an atom by number
+    /// </summary>
+    /// <param name="atomicNumber"></param>
+    /// <returns></returns>
+    /// <exception cref="Exception"></exception>
+    public static BaseAtom Resolve(int atomicNumber)
+    {
+        BaseAtom res = atomicNumber switch
+        {
+            1 => new QAtom(),
+            2 => new RAtom(),
+            3 => new EsAtom(),
+            4 => new DAtom(),
+            5 => new UeAtom(),
+            6 => new MAtom(),
+            7 => new KAtom(),
+            8 => new PAtom(),
+            9 => new SoAtom(),
+            10 => new EAtom(),
+            11 => new UAtom(),
+            12 => new AAtom(),
+            13 => new CxAtom(),
+            14 => new VAtom(),
+            15 => new ViAtom(),
+            _ => throw new Exception("TODO")
+        };
+        res.C();
+        return res;
+    }
+
+}

src/Chemistry/Bonds/Implements/BaseBond.cs

@@ -0,0 +1,318 @@
+using Skeleton.Algebra.AdditionalProperties.Extensions;
+using Skeleton.Algebra.DimensionProviders;
+using Skeleton.Algebra.Extensions;
+using Skeleton.DataStructure;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Compounds.Implements;
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+/// <summary>
+/// meta bond
+/// </summary>
+public abstract class BaseBond
+{
+
+    /// <summary>
+    /// 
+    /// </summary>
+    protected BaseBond()
+    {
+        StateMatrix = new (_ =>
+            Connected ? EigenStateMatrix * ConnectedBond.EigenStateMatrix : EigenStateMatrix
+        );
+        LogKernel = new (_ => KernelStateMatrix.Log());
+        HalfBondInformation = new(x =>
+            LogKernel.GetFrom(x) + Atom.StructureMatrix.GetFrom(x).Log()
+        );
+        StructureMatrix = new(x =>
+            (HalfBondInformation.GetFrom(x) + (Connected ? ConnectedBond.HalfBondInformation.GetFrom(x) : LieSU3.Zero))
+            .Exp()
+        );
+        EnergyDistributionFactor = new(x =>
+            {
+                C3 v = StructureMatrix.GetFrom(x).Hermitian() * ChemistryConstant.BondEnergyDistributionVector;
+                return ChemistryConstant.BondEnergyDistributionSpectrum.Rayleigh<IDim3>(v);
+            }
+        );
+        LogConnectionMatrix = new(x => Connected
+            ? (AdjointStateMatrix * ConnectedBond.AdjointStateMatrix)
+            .ConjugateOn(Atom.LogKernelMatrix.GetFrom(x))
+            : LieSU3.Zero
+        );
+        LogState = new (x => StateMatrix.GetFrom(x).Log());
+        AntibondingCatalyzeFactor = new(x =>
+            {
+                if (Connected || HomogeneousMixture == HomogeneousMixture.Null)
+                    return 0;
+                U3 s = LogState.GetFrom(x).CayleyPositive();
+                LieSU3 f0 = HomogeneousMixture.LogStateMatrix.GetFrom(x).ConjugatedBy<IDim3, LieSU3>(s);
+                LieSU3 f1 = f0 ^ LogState.GetFrom(x);
+                U3 f2 = f1.CayleyNegative();
+                return f2.Det().Imaginary;
+            }
+        );
+        BondingCatalyzeFactor = new(x =>
+            {
+                if (!Connected || HomogeneousMixture == HomogeneousMixture.Null)
+                    return 0;
+                U3 s = LogState.GetFrom(x).CayleyNegative();
+                LieSU3 f0 = HomogeneousMixture.LogStateMatrix.GetFrom(x).ConjugatedBy<IDim3, LieSU3>(s);
+                LieSU3 f1 = f0 ^ LogState.GetFrom(x);
+                U3 f2 = f1.CayleyPositive();
+                return f2.Det().Imaginary;
+            }
+        );
+
+    }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public Compound Compound => Atom.Compound;
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public HomogeneousMixture HomogeneousMixture => Compound.HomogeneousMixture; 
+    /// <summary>
+    /// null singleton
+    /// </summary>
+    public static readonly BaseBond Null = new NullBond();
+    /// <summary>
+    /// energy hold in this atom
+    /// </summary>
+    public double Energy { get; set; }
+
+    /// <summary>
+    /// how energy in the molecular is distributed into this atom
+    /// </summary>
+    public CacheItem<double> EnergyDistributionFactor { get; }
+
+    /// <summary>
+    /// the connected bond
+    /// </summary>
+    public BaseBond ConnectedBond { get; set; } = Null;
+    /// <summary>
+    /// atom that holds this bond
+    /// </summary>
+    public BaseAtom Atom { get; set; } = BaseAtom.Null;
+    /// <summary>
+    /// type of the bond
+    /// </summary>
+    public abstract string BondType { get; }
+    /// <summary>
+    /// kernel state matrix of the matrix, won't change as long as bond type is fixed
+    /// </summary>
+    public abstract SU3 KernelStateMatrix { get; }
+    /// <summary>
+    /// adj state matrix of the bond, won't change as long as bond type fixed
+    /// </summary>
+    public abstract SU3 AdjointStateMatrix { get; }
+    /// <summary>
+    /// eigen state matrix of bond, won't change as long as bond type fixed
+    /// </summary>
+    public abstract SU3 EigenStateMatrix { get; }
+
+    /// <summary>
+    /// log of the kernel state matrix
+    /// never expire
+    /// </summary>
+    public CacheItem<LieSU3> LogKernel { get; }
+
+    /// <summary>
+    /// skew hermitian matrix that represent the information of bond and its atom
+    /// </summary>
+    public CacheItem<LieSU3> HalfBondInformation { get; }
+
+    /// <summary>
+    /// special unitary matrix that represent the structure of the bond(its atom, connected bond and atom of connected bond)
+    /// </summary>
+    public CacheItem<SU3> StructureMatrix { get; }
+
+    /// <summary>
+    /// log of connection matrix
+    /// </summary>
+    public CacheItem<LieSU3> LogConnectionMatrix { get; set; }
+
+    /// <summary>
+    /// a special unitary matrix that represent the state of the bond
+    /// </summary>
+
+
+    public CacheItem<SU3> StateMatrix { get; }
+
+    /// <summary>
+    /// log of state matrix
+    /// </summary>
+    public CacheItem<LieSU3> LogState { get; }
+
+
+
+
+    /// <summary>
+    /// connect with another bond
+    /// </summary>
+    /// <param name="bond"></param>
+    /// <exception cref="Exception"></exception>
+    public void Connect(BaseBond bond)
+    {
+        if(Connected || bond.Connected)
+            return;
+        if(Atom.HomogeneousMixture != HomogeneousMixture.Null)
+            if (!Atom.Compound.Amount.IsEqualApprox(bond.Atom.Compound.Amount))
+                return;
+        if (bond == this)
+            throw new Exception("TODO");
+        ConnectedBond = bond;
+        bond.ConnectedBond = this;
+        StructureChanged();
+        bond.StructureChanged();
+        if (Atom.Compound.Atoms.Contains(bond.Atom))
+            return;
+        bond.Atom.HomogeneousMixture.Compounds.Remove(bond.Atom.Compound);
+        bond.Atom.Compound.Amount = 0d;
+        
+        foreach (BaseAtom atom in bond.Atom.Compound.Atoms.ToHashSet())
+        {
+            Atom.Compound.Atoms.Add(atom);
+            atom.Compound = Atom.Compound;
+        }
+    }
+    /// <summary>
+    /// disconnect with connected bond
+    /// </summary>
+    public void Disconnect()
+    {
+        if (!Connected)
+            return;
+        BaseBond disconnectedBond = ConnectedBond;
+        ConnectedBond = BaseBond.Null;
+        disconnectedBond.ConnectedBond = BaseBond.Null;
+        StructureChanged();
+        disconnectedBond.StructureChanged();
+        if (Atom.HasPathTo(disconnectedBond.Atom))
+            return;
+        Compound newCompound = disconnectedBond.Atom.GrabCompound;
+        newCompound.Amount = Atom.Compound.Amount;
+        
+        /*
+        newCompound.HomogeneousMixture = Atom.HomogeneousMixture;*/
+        foreach (BaseAtom atom in newCompound.Atoms)
+        {
+            atom.Compound = newCompound;
+            if (Atom.Compound.Atoms.Contains(atom))
+                Atom.Compound.Atoms.Remove(atom);
+        }
+        Atom.HomogeneousMixture.AddCompound(newCompound);
+    }
+    /// <summary>
+    /// if the bond is connected
+    /// </summary>
+    public bool Connected => ConnectedBond != Null;
+    /// <summary>
+    /// bond number, redundant information of bond type
+    /// </summary>
+    public abstract int BondNumber { get; }
+    private void StructureChanged()
+    {
+        //Atom.Compound.ResetDistanceCache();
+        Atom.Compound.CacheInit();
+        HalfBondInformation.Expire();
+        StateMatrix.Expire();
+        LogConnectionMatrix.Expire();
+        
+    }
+    
+    
+    /// <summary>
+    /// Conditions for a pair of unconnected bonds tp connect:
+    /// 1. Energy difference of the two bond is less than a threshold
+    /// 2. Energy in both bonds is higher than bonding energy
+    ///
+    /// if bond is connected, the antibonding energy is NaN
+    /// </summary>
+    /// <returns></returns>
+    public double EigenBondingEnergy()
+    {
+        if (Connected)
+            return double.NaN;
+        SU3 s1 = StructureMatrix.Get * Atom.StructureMatrix.Get;
+        C33 s2 = StructureMatrix.Get + Atom.StructureMatrix.Get;
+        double lowerLimit = -Math.PI;
+        double upperLimit = Math.PI;
+        C3 coVector = s2.Hermitian().ColumnAverage;
+        return s1
+            .ConjugateOn(new DiagR3(lowerLimit, 0d, upperLimit))
+            .Rayleigh(coVector);
+        
+    }
+    
+    /// <summary>
+    /// Conditions for a connected bond to break:
+    /// 1. energy difference of two connected bond is greater than a threshold
+    /// 2. energy in either bond is higher than antibonding energy
+    ///
+    /// if bond is unconnected, the antibonding energy is 
+    /// </summary>
+    /// <returns></returns>
+    public double EigenAntibondingEnergy()
+    {
+        if(!Connected) 
+            return double.NaN;
+        SU3 s1 = Atom.StructureMatrix.Get.Hermitian() * StructureMatrix.Get;
+        SU3 s2 = ConnectedBond.Atom.StructureMatrix.Get.Hermitian() * ConnectedBond.StructureMatrix.Get;
+        C33 sw = s1 + s2;
+        SU3 w = (s1.Log() + s2.Log()).Exp();
+        C3 coVector = sw.ColumnAverageBivariant;
+        double upperLimit = -Math.PI;
+        double lowerLimit = Math.PI;
+        return w
+            .ConjugateOn(new DiagR3(lowerLimit, 0d, upperLimit))
+            .Rayleigh(coVector);
+    }
+
+    /// <summary>
+    /// how antibonding energy affected by mixture
+    /// </summary>
+    public CacheItem<double> AntibondingCatalyzeFactor { get; set; }
+
+    /// <summary>
+    /// hot bonding energy affected by chemical environment
+    /// </summary>
+    public CacheItem<double> BondingCatalyzeFactor { get; set; }
+
+    public double BondingEnergy => EigenBondingEnergy() + BondingCatalyzeFactor.Get;
+    public double AntiBondingEnergy => EigenAntibondingEnergy() + AntibondingCatalyzeFactor.Get;
+
+    /// <summary>
+    /// how to pair bonds
+    /// </summary>
+    public int BondingGroup
+    {
+        get
+        {
+            U3 sP = HalfBondInformation.Get.CayleyPositive();
+            U3 eN = HomogeneousMixture.LogStateMatrix.Get.CayleyNegative();
+            DiagR3 h = new(0, Math.PI, Math.PI * 2);
+            U3 w = sP * eN;
+            double t = w.ConjugateOn(h).Rayleigh(sP.ColumnAverageBivariant);
+            /*
+            //Complex t = (sP * eN).Trace();// + eN.Trace() * eN.Trace();
+            C33 a1 = sP * eN;
+            C33 a2 = eN * sP;
+            C33 w = a1 - a2;
+            Complex t = (w).Det() + a1.Trace() + a2.Trace();// * (eN*sP ).Trace();// * EigenStateMatrix.Trace();*/
+            //Console.WriteLine($"TRACE  {t} -");
+
+            return t switch
+            {
+                > 7 * Math.PI / 4 or < Math.PI / 4 => 0,
+                > Math.PI / 4 and < 3 * Math.PI / 4 => 1,
+                > 3 * Math.PI / 4 and < 5 * Math.PI / 4 => 2,
+                _ => 3
+            };
+        }
+    }
+}

src/Chemistry/Bonds/Implements/DBond.cs

@@ -0,0 +1,23 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+/// <summary>
+/// bond # 3
+/// </summary>
+public class DBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.D;
+
+    /// <inheritdoc />
+    public override int BondNumber => 3;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelD;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointD;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateD;
+}

src/Chemistry/Bonds/Implements/HBond.cs

@@ -0,0 +1,24 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 7
+/// </summary>
+public class HBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.H;
+
+    /// <inheritdoc />
+    public override int BondNumber => 7;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelH;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointH;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateH;
+}

src/Chemistry/Bonds/Implements/MBond.cs

@@ -0,0 +1,25 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 1
+/// </summary>
+public class MBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.M;
+
+    /// <inheritdoc />
+    public override int BondNumber => 1;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelM;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointM;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateM;
+
+}

src/Chemistry/Bonds/Implements/NullBond.cs

@@ -0,0 +1,27 @@
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond singleton that represent no bond
+/// </summary>
+public class NullBond : BaseBond
+{
+    /// <inheritdoc />
+    public NullBond()
+    {
+    }
+
+    /// <inheritdoc />
+    public override string BondType => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => throw new Exception();
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => throw new Exception();
+
+    /// <inheritdoc />
+    public override int BondNumber => throw new Exception();
+}

src/Chemistry/Bonds/Implements/PBond.cs

@@ -0,0 +1,24 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 6
+/// </summary>
+public class PBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.P;
+
+    /// <inheritdoc />
+    public override int BondNumber => 6;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelP;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointP;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateP;
+}

src/Chemistry/Bonds/Implements/QBond.cs

@@ -0,0 +1,25 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 5
+/// </summary>
+public class QBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.Q;
+
+    /// <inheritdoc />
+    public override int BondNumber => 5;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelQ;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointQ;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateQ;
+    
+}

src/Chemistry/Bonds/Implements/SBond.cs

@@ -0,0 +1,24 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 2
+/// </summary>
+public class SBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.S;
+
+    /// <inheritdoc />
+    public override int BondNumber => 2;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelS;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointS;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateS;
+}

src/Chemistry/Bonds/Implements/YBond.cs

@@ -0,0 +1,24 @@
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Implements;
+
+/// <summary>
+/// bond # 4
+/// </summary>
+public class YBond : BaseBond
+{
+    /// <inheritdoc />
+    public override string BondType => ChemistryConstant.BondTypes.Y;
+
+    /// <inheritdoc />
+    public override int BondNumber => 4;
+
+    /// <inheritdoc />
+    public override SU3 KernelStateMatrix => ChemistryConstant.BondKernels.BondKernelY;
+
+    /// <inheritdoc />
+    public override SU3 AdjointStateMatrix => ChemistryConstant.BondAdjoints.BondAdjointY;
+
+    /// <inheritdoc />
+    public override SU3 EigenStateMatrix => ChemistryConstant.BondEigenStates.BondEigenStateY;
+}

src/Chemistry/Bonds/Resolver/BondResolver.cs

@@ -0,0 +1,53 @@
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Constants;
+
+namespace VirtualChemistry.Chemistry.Bonds.Resolver;
+
+/// <summary>
+/// create a bond instance by descriptions 
+/// </summary>
+public static class BondResolver
+{
+    /// <summary>
+    /// create the bond by type
+    /// </summary>
+    /// <param name="type"></param>
+    /// <returns></returns>
+    /// <exception cref="Exception"></exception>
+    public static BaseBond Resolve(string type)
+    {
+        BaseBond res = type switch
+        {
+            ChemistryConstant.BondTypes.M => new MBond(),
+            ChemistryConstant.BondTypes.S => new SBond(),
+            ChemistryConstant.BondTypes.D => new DBond(),
+            ChemistryConstant.BondTypes.Y => new YBond(),
+            ChemistryConstant.BondTypes.Q => new QBond(),
+            ChemistryConstant.BondTypes.P => new PBond(),
+            ChemistryConstant.BondTypes.H => new HBond(),
+            _ => throw new Exception("TODO")
+        };
+        return res;
+    }
+    /// <summary>
+    /// create a bond by number
+    /// </summary>
+    /// <param name="number"></param>
+    /// <returns></returns>
+    /// <exception cref="Exception"></exception>
+    public static BaseBond Resolve(int number)
+    {
+        BaseBond res = number switch
+        {
+            1 => new MBond(),
+            2 => new SBond(),
+            3 => new DBond(),
+            4 => new YBond(),
+            5 => new QBond(),
+            6 => new PBond(),
+            7 => new HBond(),
+            _ => throw new Exception("TODO")
+        };
+        return res;
+    }
+}

src/Chemistry/Compounds/Implements/Compound.cs

@@ -0,0 +1,477 @@
+using Skeleton.Algebra.Extensions;
+using Skeleton.DataStructure;
+using Skeleton.DataStructure.Packs;
+using Skeleton.Utils;
+using Skeleton.Utils.Helpers;
+using Skeleton.Utils.InverseSampling;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Chemistry.Compounds.Resolver;
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+using VirtualChemistry.DataStructure.Packs;
+
+namespace VirtualChemistry.Chemistry.Compounds.Implements;
+
+/// <summary>
+/// a cluster of molecular of the same kind
+/// </summary>
+public class Compound
+{
+    /// <summary>
+    /// singleton used to represent no compound
+    /// </summary>
+    public static readonly Compound Null = new Compound();
+    private string? IsoRepresentationCache { get; set; }
+    /// <summary>
+    /// Constructor
+    /// </summary>
+    public Compound()
+    {
+        CacheInit();
+        LogStateMatrix = new(x => Atoms
+            .Select(atom => atom.LogStateMatrix.GetFrom(x))
+            .DefaultIfEmpty(LieSU3.Zero)
+            .SpLieSum()
+        );
+        StateMatrix = new(x => LogStateMatrix.GetFrom(x).Exp());
+        CompressionElasticity = new (x => (
+                LogStateMatrix.GetFrom(x).CayleyNegative() * StateMatrix.GetFrom(x))
+            .ConjugateOn(new DiagR3(0d, 2d * Math.PI, 4d * Math.PI))
+            .Rayleigh(LogStateMatrix.GetFrom(x).ColumnAverage)
+        );
+        CompressionBias = new (x =>
+            {
+                DiagR3 w1 = new (1d / 11d, 7d / 11d, 10d / 11d);
+                DiagR3 w2 = new (2d / 17d, 8d / 17d, 16d / 17d);
+                double p1 = StateMatrix.GetFrom(x).ConjugateOn(w1).Rayleigh(StateMatrix.GetFrom(x).Hermitian().ColumnAverage);
+                double p2 = StateMatrix.GetFrom(x).Hermitian().ConjugateOn(w2).Rayleigh(StateMatrix.GetFrom(x).ColumnAverageBivariant);
+                double x1 = InverseSampling.StandardNormal(p1) / 4d - 2d * Math.PI;
+                double x2 = InverseSampling.StandardNormal(p2) / 3d + 2d * Math.PI;
+                return Math.Atan(Math.Abs(x1) > Math.Abs(x2) ? x1 : x2) * 2d;
+            }
+        );
+        FreeDensity = new(x =>
+        {
+            double avg = (Math.Exp(-4d * Math.PI) + Math.Exp(Math.PI)) / 2d;
+            DiagR3 spectrum = new DiagR3(Math.Exp(-4d * Math.PI), avg, Math.Exp(Math.PI));
+            double tempMod = Math.Exp(-Phase);
+            return tempMod * LogStateMatrix.GetFrom(x)
+                .CayleyPositive()
+                .ConjugateOn(spectrum)
+                .Rayleigh(StateMatrix.GetFrom(x).ColumnAverageBivariant);
+        });
+    }
+    /// <summary>
+    /// initialize cache
+    /// </summary>
+    public void CacheInit() 
+    {
+        HomogeneousMixture.ResetAllCache();
+        ResetDistanceCache();
+        IsoRepresentationCache = null;
+    }
+    /// <summary>
+    /// 
+    /// </summary>
+    public List<string> TraceLog { get; } = new();
+    /// <summary>
+    /// compression elasticity, refer to the homogeneous mixture version
+    /// </summary>
+    public CacheItem<double> CompressionElasticity { get; }
+    /// <summary>
+    /// compression bias, refer to the homogeneous mixture version
+    /// </summary>
+    public CacheItem<double> CompressionBias { get; }
+    /// <summary>
+    /// all atoms in the molecular
+    /// </summary>
+    public HashSet<BaseAtom> Atoms { get; set; } = new ();
+    /// <summary>
+    /// all connections in this compound
+    /// </summary>
+    public IOrderedEnumerable<FullConnectionInfo> Connections
+    {
+        get
+        {
+            HashSet<BaseBond> visited = new ();
+            List<FullConnectionInfo> res = new ();
+            foreach (BaseBond bond in Bonds.Where(bond => bond.Connected))
+            {
+                if(visited.Contains(bond))
+                    continue;
+                if(res.All(con => !con.TryAbsorbBond(bond)))
+                    res.Add(new FullConnectionInfo(bond));
+                visited.Add(bond);
+                visited.Add(bond.ConnectedBond);
+            }
+            return res.OrderBy(con => con.Representation);
+        }
+    }
+    
+
+    public IEnumerable<BaseBond> UnconnectedIndex() => Atoms.Count > 15 ?
+        Array.Empty<BaseBond>():
+        Bonds.Where(bond => !bond.Connected && bond.Energy > bond.BondingEnergy);
+    
+
+    /// <summary>
+    /// string used to save the compound
+    /// </summary>
+    public string ConnectionInfo => String.Join(':', Connections.Select(con => con.Representation));
+    /// <summary>
+    /// all bonds in the compound
+    /// </summary>
+    public IEnumerable<BaseBond> Bonds => Atoms.SelectMany(atom => atom.Bonds);
+    /// <summary>
+    /// the homogeneous mixture of this compound
+    /// </summary>
+    public HomogeneousMixture HomogeneousMixture { get; set; } = HomogeneousMixture.Null;
+    /// <summary>
+    /// state matrix of the compound
+    /// </summary>
+    public CacheItem<SU3> StateMatrix { get; }
+
+    /// <summary>
+    /// log state matrix
+    /// </summary>
+    public CacheItem<LieSU3> LogStateMatrix { get; }
+
+    /// <summary>
+    /// a quantity to compare phase among different materials
+    /// </summary>
+    public double Phase => Math.Tanh(Temperature * CompressionElasticity.Get + CompressionBias.Get);
+
+    /// <summary>
+    /// compression stiffness, refer to the homogeneous mixture version
+    /// </summary>
+    public double CompressionStiffness => Math.Exp(Math.PI / 2d * (5d * Phase - 3d));
+    
+    /// <summary>
+    /// temperature of the compound
+    /// </summary>
+    public double Temperature => HomogeneousMixture.Temperature;
+    
+    /// <summary>
+    /// unique chemical expression of the compound
+    /// </summary>
+    public string Expression =>
+        Atoms
+            .GroupBy(atom => atom.Element)
+            .OrderBy(group => group.First().AtomicNumber)
+            .Select(group => $"{group.First().Element}"+ (group.Count() > 1 ? group.Count().ToString() : "") )
+            .DefaultIfEmpty("")
+            .Aggregate((a, b) => a + b);
+    /// <summary>
+    /// dump the compound into save string
+    /// </summary>
+    /// <returns></returns>
+    /// <exception cref="Exception"></exception>
+    public string Dump()
+    {
+        Dictionary<BaseAtom, string> atomMap = new Dictionary<BaseAtom, string>();
+        Dictionary<BaseBond, string> bondMap = new Dictionary<BaseBond, string>();
+        HashSet<string> records = new HashSet<string>();
+        foreach (BaseAtom atom in Atoms)
+            atomMap[atom] = $"({atom.Element}.{atomMap.Count})";
+        foreach (BaseBond bond in Bonds.Where(bond => bond.Connected))
+            bondMap[bond] = $"<{bond.BondType}.{bondMap.Count}>";
+        
+        foreach (BaseAtom atom in Atoms)
+        {
+            if (atom.Bonds.All(bond => !bond.Connected))
+            {
+                records.Add($"{atomMap[atom]}-<xxx.0>-<xxx.0>-(xxx.0)");
+                continue;
+            }
+
+            foreach (BaseBond bond in atom.Bonds.Where(bond => bond.Connected))
+            {
+                if (bond == bond.ConnectedBond)
+                    throw new Exception("TODO");
+                if (bondMap[bond] == bondMap[bond.ConnectedBond])
+                    throw new Exception("TODO");
+                string record =
+                    $"{atomMap[atom]}-{bondMap[bond]}-{bondMap[bond.ConnectedBond]}-{atomMap[bond.ConnectedBond.Atom]}";
+                string revRecord =
+                    $"{atomMap[bond.ConnectedBond.Atom]}-{bondMap[bond.ConnectedBond]}-{bondMap[bond]}-{atomMap[atom]}";
+                if(records.Contains(revRecord))
+                    continue;
+                records.Add(record);
+
+            }
+        }
+        return String.Join('|', records) + $"<CompAmtSp>{Amount.ExactDoubleString()}";
+    }
+    /// <summary>
+    /// dump with mapped infomation
+    /// </summary>
+    /// <returns></returns>
+    public CompoundDumpMap MappedDump()
+    {
+        Dictionary<BaseAtom, string> atomMap = new();
+        Dictionary<BaseBond, string> bondMap = new();
+        HashSet<string> records = new();
+            foreach (BaseAtom atom in Atoms)
+                atomMap[atom] = $"{atom.Element}.{atomMap.Count}";
+        foreach (BaseBond bond in Bonds.Where(bond => bond.Connected))
+            bondMap[bond] = $"{bond.BondType}.{bondMap.Count}";
+            
+        foreach (BaseAtom atom in Atoms)
+        {
+            if (atom.Bonds.All(bond => !bond.Connected))
+            {
+                records.Add($"({atomMap[atom]})-<xxx.0>-<xxx.0>-(xxx.0)");
+                continue;
+            }
+
+            foreach (BaseBond bond in atom.Bonds.Where(bond => bond.Connected))
+            {
+                string record =
+                    $"({atomMap[atom]})-<{bondMap[bond]}>-<{bondMap[bond.ConnectedBond]}>-({atomMap[bond.ConnectedBond.Atom]})";
+                string revRecord =
+                    $"({atomMap[bond.ConnectedBond.Atom]})-<{bondMap[bond.ConnectedBond]}>-<{bondMap[bond]}>-({atomMap[atom]})";
+                if(records.Contains(revRecord))
+                    continue;
+                records.Add(record);
+            }
+        }
+        string dumpString = String.Join('|', records);
+        return new CompoundDumpMap(atomMap, bondMap, dumpString);
+    }
+    
+    /// <summary>
+    /// make a copy of the compound
+    /// </summary>
+    /// <returns></returns>
+    public Compound Copy() => CompoundResolver.Resolve(Dump());
+
+    /// <summary>
+    /// make a copy with map
+    /// </summary>
+    /// <param name="handle"></param>
+    /// <returns></returns>
+    public PlainPack2<Compound, BaseBond> CopyWithBondMap(BaseBond handle)
+    {
+        CompoundDumpMap dm = MappedDump();
+        CompoundResolveMap rm = CompoundResolver.MappedResolve(dm.DumpString);
+        if (dm.BondMap.ContainsKey(handle))
+            return new PlainPack2<Compound, BaseBond>(rm.ResolvedCompound, rm.BondMap[dm.BondMap[handle]]);
+        return new PlainPack2<Compound, BaseBond>(rm.ResolvedCompound, rm.AtomMap[dm.AtomMap[handle.Atom]].NextUnconnected(handle.BondType));
+    }
+    /// <summary>
+    /// distance between atoms
+    /// </summary>
+    public Dictionary<BaseAtom, Dictionary<BaseAtom, int>> DistanceCache { get; set; } =
+        new Dictionary<BaseAtom, Dictionary<BaseAtom, int>>();
+    /// <summary>
+    /// get the cached distance between two atoms in the compound
+    /// </summary>
+    /// <param name="a1"></param>
+    /// <param name="a2"></param>
+    /// <returns></returns>
+    public int CachedDistance(BaseAtom a1, BaseAtom a2)
+    {
+        if(DistanceCache.Keys.Contains(a1))
+            if (DistanceCache[a1].Keys.Contains(a2))
+                return DistanceCache[a1][a2];
+        if(DistanceCache.Keys.Contains(a2))
+            if (DistanceCache[a2].Keys.Contains(a1))
+                return DistanceCache[a2][a1];
+        return -1;
+    }
+    /// <summary>
+    /// save distance of two atoms in this compound into cache
+    /// </summary>
+    /// <param name="a1"></param>
+    /// <param name="a2"></param>
+    /// <param name="distance"></param>
+    public void CacheDistance(BaseAtom a1, BaseAtom a2, int distance)
+    {
+        if (!DistanceCache.ContainsKey(a1))
+            DistanceCache[a1] = new Dictionary<BaseAtom, int>();
+        DistanceCache[a1][a2] = distance;
+    }
+    /// <summary>
+    /// reset the cache
+    /// </summary>
+    public void ResetDistanceCache() => DistanceCache = new Dictionary<BaseAtom, Dictionary<BaseAtom, int>>();
+    /// <summary>
+    /// if two compound has the same structure
+    /// </summary>
+    /// <param name="compound"></param>
+    /// <returns></returns>
+    public bool IsometricTo(Compound compound)
+    {
+        if (!Expression.Equals(compound.Expression))
+            return false;
+        if(!ConnectionInfo.Equals(compound.ConnectionInfo))
+            return false;
+        if (!StateMatrix.Get.IsEqualApprox(compound.StateMatrix.Get))
+            return false;
+        return IsoRepresentation.Equals(compound.IsoRepresentation, StringComparison.OrdinalIgnoreCase);
+    }
+    /// <summary>
+    /// amount of molecular in this compound
+    /// </summary>
+    public double Amount { get; set; }
+    /// <summary>
+    /// energy per amount
+    /// </summary>
+    public double UnitEnergy
+    {
+        get => Energy / Amount;
+        set => Energy = value * Amount;
+    }
+    /// <summary>
+    /// energy
+    /// </summary>
+    public double Energy
+    {
+        get => Atoms.Sum(atom => atom.Energy) * Amount;
+        set
+        {
+            double energyPerAmount = value / Amount;
+            double energyFactor = Atoms.Sum(atom => atom.EnergyDistributionFactor.Get);
+            foreach (BaseAtom atom in Atoms)
+                atom.Energy = energyPerAmount * atom.EnergyDistributionFactor.Get / energyFactor;
+        }
+    }
+    /// <summary>
+    /// amount : total mixture amount
+    /// </summary>
+    public double Concentration =>
+        Amount / (HomogeneousMixture == HomogeneousMixture.Null ? Amount : HomogeneousMixture.Amount);
+    /// <summary>
+    /// split into two compounds
+    /// </summary>
+    /// <param name="amount"></param>
+    /// <param name="byRatio"></param>
+    /// <returns></returns>
+    public Compound Split(double amount, bool byRatio = false)
+    {
+        if (amount.IsEqualApprox(0))
+            return Null;
+        if ((!byRatio && amount.IsEqualApprox(Amount)) || (byRatio && amount.IsEqualApprox(1)))
+        {
+            HomogeneousMixture.Compounds.Remove(this);
+            HomogeneousMixture = HomogeneousMixture.Null;
+            return this;
+        }
+        double ebs = Energy;
+        Compound res = Copy();
+        if(!byRatio)
+        {
+            double ratio = amount / Amount;
+            Amount -= amount;
+            res.Amount = amount;
+            res.Energy = ebs * ratio;
+            Energy = ebs * (1d - ratio);
+        }
+        else
+        {
+            res.Amount = Amount * amount;
+            Amount *= (1d - amount);
+            res.Energy = ebs * amount;
+            Energy = ebs * (1d - amount);
+        }
+        res.HomogeneousMixture = HomogeneousMixture.Null;
+        return res;
+    }
+    /// <summary>
+    /// split with bond information
+    /// </summary>
+    /// <param name="amount"></param>
+    /// <param name="handle"></param>
+    /// <param name="byRatio"></param>
+    /// <returns></returns>
+    public PlainPack2<Compound, BaseBond> SplitWithBondMap(double amount, BaseBond handle, bool byRatio = false)
+    {
+        /*if (Amount.IsEqualApprox(0))
+            throw new Exception("DD");
+        if ((Amount * 0.5).IsEqualApprox(0))
+            throw new Exception("FS");*/
+        PlainPack2<Compound, BaseBond> res = CopyWithBondMap(handle);
+        if (!byRatio)
+        {
+            double ratio = amount / Amount;
+            double splitEnergy = Energy * ratio;
+            double energy = Energy - splitEnergy;
+            Amount -= amount;
+            res.Item1.Amount = amount;
+            Energy = energy;
+            res.Item1.Energy = splitEnergy;
+        }
+        else
+        {
+            double splitEnergy = Energy * amount;
+            double energy = Energy * (1 - amount);
+            res.Item1.Amount = Amount * amount;
+            Amount *= 1 - amount;
+            Energy = energy;
+            res.Item1.Energy = splitEnergy;
+        }
+        HomogeneousMixture.AddCompound(res.Item1, true, true);
+        return res;
+    }
+    /// <summary>
+    /// mass per molecular
+    /// </summary>
+    public double MolecularMass => Atoms
+        .Select(atom => atom.AtomicMass)
+        .DefaultIfEmpty(0d)
+        .Sum();
+    /// <summary>
+    /// unique representation of a compound
+    /// </summary>
+    public string IsoRepresentation => IsoRepresentationCache ??= Atoms
+        .GroupedMinBy(a => -a.AtomicNumber)
+        .GroupedMinBy(a => a.ConnectedAtoms.Count())
+        .GroupedApproxMinBy(a => a.GroupMass)
+        .GroupedApproxMinBy(a => a.StateMatrix.Get.Det().Real)
+        .GroupedApproxMinBy(a => a.StateMatrix.Get.Trace().Real)
+        .GroupedApproxMinBy(a => a.StructureMatrix.Get.Det().Real)
+        .Select(a => a.FullRepresentation)
+        .Min() ?? "";
+
+
+    /// <summary>
+    /// a real number from 0 to 1
+    /// depend on temperature
+    /// Resolve Level
+    /// when A is next to B(above or below)
+    /// the one with higher resolve level will try to absorb the other one
+    /// 
+    /// Higher Level => behaviour like solvent
+    /// Lower Level => behaviour like solute 
+    /// </summary>
+    public double ResolveLevel
+    {
+        get
+        {
+            double Normalizer(double x) => 2d * (Math.Tanh(-x * 3d) + 1d) / 5d + 1d / 5d;
+            double C(double x) => 1 - Math.Exp(-x);
+            double z = C(C(Phase));
+            C3 divisor = (StateMatrix.Get.ColumnAverage + LogStateMatrix.Get.ColumnAverageBivariant.Conj()) / 2d;
+            double factor = StateMatrix.Get.ConjugateOn(new DiagR3(1d, 1d / 9d, 1d / 3d)).Rayleigh(divisor);
+            return factor * Normalizer(Phase) / Math.Cosh(z);
+        }
+    }
+    
+    
+    /*
+     * Phase of compound is determined by compression stiffness
+     * higher stiffness => more like gas
+     * lower stiffness => more like solid
+     *
+     * resolvability of one solute A in solvent B(resolve level of A is less than B)
+     * if both A and B are solid => 0
+     *
+     * otherwise determined by many factors
+     *
+     */
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public CacheItem<double> FreeDensity { get; }
+}

src/Chemistry/Compounds/Resolver/CompoundResolver.cs

@@ -0,0 +1,111 @@
+using System.Text.RegularExpressions;
+using Skeleton.Utils;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Atoms.Resolver;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Chemistry.Compounds.Implements;
+using VirtualChemistry.DataStructure.Packs;
+
+namespace VirtualChemistry.Chemistry.Compounds.Resolver;
+
+/// <summary>
+/// get compound by save string
+/// </summary>
+public static class CompoundResolver
+{
+    private const string AtomRegex = @"\(([a-zA-Z]+\.[0-9]+)\)";
+    private const string BondRegex = @"<([a-zA-Z]+\.[0-9]+)>";
+
+
+    /// <summary>
+    /// resolve into map
+    /// </summary>
+    /// <param name="expression"></param>
+    /// <returns></returns>
+    public static CompoundResolveMap MappedResolve(string expression)
+    {
+        Compound res = new Compound();
+        IEnumerable<string> records = expression.Split('|');
+        Dictionary<string, BaseAtom> atomMap = new();
+        Dictionary<string, BaseBond> bondMap = new();
+        foreach (string record in records)
+        {
+            List<string> connection = record.Split('-').ToList();
+            string atom1 = new Regex(AtomRegex).Match(connection[0]).Result("$1");
+            string bond1 = new Regex(BondRegex).Match(connection[1]).Result("$1");
+            string bond2 = new Regex(BondRegex).Match(connection[2]).Result("$1");
+            string atom2 = new Regex(AtomRegex).Match(connection[3]).Result("$1");
+            if (bond1.Equals("xxx.0", StringComparison.OrdinalIgnoreCase))
+            {
+                BaseAtom single = AtomResolver.Resolve(atom1.Split('.')[0]);
+                atomMap[atom1] = single;
+                single.Compound = res;
+                res.Atoms.Add(single);
+                res.CacheInit();
+                return new CompoundResolveMap(atomMap, bondMap, res);
+            }
+
+            if (!atomMap.Keys.Contains(atom1))
+                atomMap[atom1] = AtomResolver.Resolve(atom1.Split('.')[0]);
+            if (!atomMap.Keys.Contains(atom2))
+                atomMap[atom2] = AtomResolver.Resolve(atom2.Split('.')[0]);
+            atomMap[atom1].Compound = res;
+            atomMap[atom2].Compound = res;
+            res.Atoms.Add(atomMap[atom1]);
+            res.Atoms.Add(atomMap[atom2]);
+            BaseBond ba = atomMap[atom1].NextUnconnected(bond1.Split('.')[0]);
+            BaseBond bb = atomMap[atom2].NextUnconnected(bond2.Split('.')[0], ba);
+
+            ba.Connect(bb);
+            bondMap[bond1] = ba;
+            bondMap[bond2] = bb;
+        }
+        res.CacheInit();
+        return new CompoundResolveMap(atomMap, bondMap, res);
+    }
+    /// <summary>
+    /// get the compound by save string
+    /// </summary>
+    /// <param name="expression"></param>
+    /// <returns></returns>
+    public static Compound Resolve(string expression)
+    {
+        string[] parts = expression.Split("<CompAmtSp>");
+        expression = parts[0];
+        Compound res = new Compound();
+        res.Amount = parts[1].ByteStringToDouble();
+        IEnumerable<string> records = expression.Split('|');
+        Dictionary<string, BaseAtom> atomMap = new Dictionary<string, BaseAtom>();
+        foreach (string record in records)
+        {
+            List<string> connection = record.Split('-').ToList();
+            string atom1 = new Regex(AtomRegex).Match(connection[0]).Result("$1");
+            string bond1 = new Regex(BondRegex).Match(connection[1]).Result("$1");
+            string bond2 = new Regex(BondRegex).Match(connection[2]).Result("$1");
+            string atom2 = new Regex(AtomRegex).Match(connection[3]).Result("$1");
+            if (bond1.Equals("xxx.0", StringComparison.OrdinalIgnoreCase))
+            {
+                BaseAtom single = AtomResolver.Resolve(atom1.Split('.')[0]);
+                single.Compound = res;
+                res.Atoms.Add(single);
+                res.CacheInit();
+                return res;
+            }
+
+            if (!atomMap.Keys.Contains(atom1))
+                atomMap[atom1] = AtomResolver.Resolve(atom1.Split('.')[0]);
+            if (!atomMap.Keys.Contains(atom2))
+                atomMap[atom2] = AtomResolver.Resolve(atom2.Split('.')[0]);
+            atomMap[atom1].Compound = res;
+            atomMap[atom2].Compound = res;
+            res.Atoms.Add(atomMap[atom1]);
+            res.Atoms.Add(atomMap[atom2]);
+            BaseBond b1 = atomMap[atom1].NextUnconnected(bond1.Split('.')[0]);
+            BaseBond b2 = atomMap[atom2].NextUnconnected(bond2.Split('.')[0], b1);
+            b1.Connect(b2);
+        }
+        res.CacheInit();
+        return res;
+    }
+
+}

src/Chemistry/Containers/IChemicalContainer.cs

@@ -0,0 +1,27 @@
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+
+namespace VirtualChemistry.Chemistry.Containers;
+
+/// <summary>
+/// 
+/// </summary>
+public interface IChemicalContainer
+{
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <returns></returns>
+    double Volume();
+    /// <summary>
+    /// 
+    /// </summary>
+    HeterogeneousMixture Content { get; set; }
+    /// <summary>
+    /// 
+    /// </summary>
+    public double EnvironmentPressure { get; set; }
+    /// <summary>
+    /// 
+    /// </summary>
+    public double EnvironmentTemperature { get; set; }
+}

src/Chemistry/Containers/NullContainer.cs

@@ -0,0 +1,29 @@
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+
+namespace VirtualChemistry.Chemistry.Containers;
+/// <summary>
+/// container singleton used to represent no container
+/// </summary>
+public sealed class NullContainer : IChemicalContainer
+{
+    private NullContainer()
+    {
+    }
+
+    /// <summary>
+    /// instance
+    /// </summary>
+    public static readonly NullContainer Null = new();
+
+    /// <inheritdoc />
+    public double Volume() => Double.PositiveInfinity;
+
+    /// <inheritdoc />
+    public HeterogeneousMixture Content { get; set; } = HeterogeneousMixture.Null;
+
+    /// <inheritdoc />
+    public double EnvironmentPressure { get; set; }
+
+    /// <inheritdoc />
+    public double EnvironmentTemperature { get; set; }
+}

src/Chemistry/Environments/Implements/Environment.cs

@@ -0,0 +1,12 @@
+namespace VirtualChemistry.Chemistry.Environments.Implements;
+
+/// <summary>
+/// the environment
+/// </summary>
+public class Environment
+{
+    /// <summary>
+    /// environment temperature
+    /// </summary>
+    public double Temperature { get; set; }
+}

src/Chemistry/Mixtures/Implements/HeterogeneousMixture.cs

@@ -0,0 +1,158 @@
+using Skeleton.DataStructure.Link;
+using VirtualChemistry.Chemistry.Containers;
+using VirtualChemistry.Chemistry.Mixtures.Resolver;
+
+namespace VirtualChemistry.Chemistry.Mixtures.Implements;
+
+/// <summary>
+/// a collection of homogeneous mixtures 
+/// </summary>
+public class HeterogeneousMixture
+{
+    public HeterogeneousMixture()
+    {
+    }
+
+    public HeterogeneousMixture(IChemicalContainer c)
+    {
+        Container = c;
+    }
+
+    /// <summary>
+    /// singleton used to represent no mixture
+    /// </summary>
+    public static readonly HeterogeneousMixture Null = new HeterogeneousMixture();
+    /// <summary>
+    /// container of the mixture
+    /// </summary>
+    public IChemicalContainer Container { get; set; } = NullContainer.Null;
+    /// <summary>
+    /// volume of the container
+    /// </summary>
+    public double ContainerVolume => Container.Volume();
+    /// <summary>
+    /// all homogeneous mixtures within
+    /// </summary>
+    public HashSet<HomogeneousMixture> Layers { get; set; } = new();
+    /// <summary>
+    /// homogeneous mixtures in order from bottom to top
+    /// </summary>
+    public Link<HomogeneousMixture> LayerOrder { get; set; } = new();
+    /// <summary>
+    /// add an empty homo mixture to the layers
+    /// </summary>
+    /// <returns></returns>
+    public HomogeneousMixture AddLayer()
+    {
+        HomogeneousMixture res = new HomogeneousMixture();
+        AddLayer(res);
+        return res;
+    }
+
+    /// <summary>
+    /// remove a homo from the layers
+    /// </summary>
+    /// <param name="m"></param>
+    public void RemoveLayer(HomogeneousMixture m)
+    {
+        if (this == Null)
+            return;
+        Layers.Remove(m);
+        LayerOrder.Remove(m.Layer);
+        m.HeterogeneousMixture = Null;
+        m.Layer = LinkNode<HomogeneousMixture>.Null;
+    }
+
+    /// <summary>
+    /// add a homo mixture into this as a new layer
+    /// </summary>
+    /// <param name="m"></param>
+    /// <param name="atHead"></param>
+    public void AddLayer(HomogeneousMixture m, bool atHead=true)
+    {
+        m.HeterogeneousMixture.RemoveLayer(m);
+        m.HeterogeneousMixture = this;
+        Layers.Add(m);
+        if (atHead)
+            LayerOrder.AddFirst(m);
+        else
+            LayerOrder.AddLast(m);
+    }
+    /// <summary>
+    /// total amount of molecular
+    /// </summary>
+    public double Amount
+    {
+        get => Layers
+            .Select(l => l.Amount)
+            .DefaultIfEmpty(0d)
+            .Sum();
+        set
+        {
+            Dictionary<HomogeneousMixture, double> res = new Dictionary<HomogeneousMixture, double>();
+            foreach (HomogeneousMixture l in Layers)
+                res[l] = value * l.Ratio;
+            foreach (HomogeneousMixture l in Layers)
+                l.Amount = res[l];
+        }
+    }
+    /// <summary>
+    /// if this degenerate into a single homo mixture
+    /// </summary>
+    public bool IsHomogeneous => Layers.Count == 1;
+
+    /// <summary>
+    /// dump the mixture into save string
+    /// </summary>
+    /// <returns></returns>
+    public string Dump() =>
+        $"`LAYERS{String.Join(HeterogeneousMixtureResolver.LayerSplit, LayerOrder.Select(layer => layer.Dump()))}`%LAYERS";
+
+    /// <summary>
+    /// volume of contents
+    /// </summary>
+    public double Volume => Layers.Sum(layer => layer.Volume);
+
+    /// <summary>
+    /// the total stiffness of all contents
+    /// </summary>
+    public double EquivalentStiffness => 1d / Layers.Sum(h => 1d / h.CompressionStiffness);
+
+    /// <summary>
+    /// volume of contents under 0 pressure
+    /// </summary>
+    public double FreeVolume => Layers.Sum(h => h.FreeVolume);
+
+    private double ContentVolume => Layers.Sum(h => h.Volume);
+    /// <summary>
+    /// the force from container
+    /// </summary>
+    public double ForceFromContainer => (Math.Min(FreeVolume, Volume) - ContentVolume) * EquivalentStiffness;
+
+    /// <summary>
+    /// process one step of all its homogeneous mixture<br/>
+    /// including reaction, resolving/precipitating compounds,<br/>
+    /// heat exchange and volume update 
+    /// </summary>
+    public void OneStep()
+    {
+        foreach (HomogeneousMixture h in Layers)
+        {
+            if(h.Layer.Next.IsEnding)
+                continue;
+            if(h.Density > h.Layer.Next.Value.Density)
+                h.Layer.MoveForward();
+        }
+        foreach (HomogeneousMixture h in Layers)
+            h.React();
+        foreach (HomogeneousMixture h in Layers.ToArray())
+            h.PrecipitateAndResolve();
+        foreach (HomogeneousMixture h in Layers)
+        {
+            h.Degenerate();
+            h.VolumeUpdate();
+            h.HeatExchange();
+        }
+    }
+}
+

src/Chemistry/Mixtures/Implements/HomogeneousMixture.cs

@@ -0,0 +1,772 @@
+using System.Numerics;
+using Skeleton.Algebra.Extensions;
+using Skeleton.Algebra.ScalarFieldStructure;
+using Skeleton.DataStructure;
+using Skeleton.DataStructure.Link;
+using Skeleton.DataStructure.Packs;
+using Skeleton.Utils;
+using Skeleton.Utils.Helpers;
+using Skeleton.Utils.InverseSampling;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Chemistry.Compounds.Implements;
+using VirtualChemistry.Chemistry.Mixtures.Resolver;
+using VirtualChemistry.Constants;
+using VirtualChemistry.DataStructure.Packs;
+
+namespace VirtualChemistry.Chemistry.Mixtures.Implements;
+/// <summary>
+/// mixture of many well mixed compounds that can not be seperated by psy methods
+/// </summary>
+public class HomogeneousMixture
+{
+    /// <summary>
+    /// Constructor by default
+    /// </summary>
+    public HomogeneousMixture()
+    {
+        LogStateMatrix = new CacheItem<LieSU3>(x =>
+            Compounds.Select(compound => compound.LogStateMatrix.GetFrom(x) * compound.Concentration)
+                .DefaultIfEmpty(LieSU3.Zero)
+                .SpLieSum()
+        );
+        StateMatrix = new(x => LogStateMatrix.GetFrom(x).Exp());
+        CompressionElasticity = new(x =>
+            {
+                DiagR3 spectrum = new (0d, 2d * Math.PI, 4d * Math.PI);
+                return (LogStateMatrix.GetFrom(x).CayleyNegative() * StateMatrix.GetFrom(x))
+                    .ConjugateOn(spectrum)
+                    .Rayleigh(LogStateMatrix.GetFrom(x).ColumnAverage);
+            }
+        );
+        CompressionBias = new(x =>
+            {
+                DiagR3 w1 = new (1d / 11d, 7d / 11d, 10d / 11d);
+                DiagR3 w2 = new (2d / 17d, 8d / 17d, 16d / 17d);
+                double p1 = StateMatrix.GetFrom(x)
+                    .ConjugateOn(w1)
+                    .Rayleigh(StateMatrix.GetFrom(x).Hermitian().ColumnAverage);
+                double p2 = StateMatrix.GetFrom(x)
+                    .Hermitian()
+                    .ConjugateOn(w2)
+                    .Rayleigh(StateMatrix.GetFrom(x).ColumnAverageBivariant);
+                double x1 = InverseSampling.StandardNormal(p1) / 4d - 2d * Math.PI;
+                double x2 = InverseSampling.StandardNormal(p2) / 3d + 2d * Math.PI;
+                return Math.Atan(Math.Abs(x1) > Math.Abs(x2) ? x1 : x2) * 2d;
+            }
+        );
+        FreeDensity = new(x =>
+            {
+                double avg = (Math.Exp(-4d * Math.PI) + Math.Exp(Math.PI)) / 2d;
+                DiagR3 spectrum = new DiagR3(Math.Exp(-4d * Math.PI), avg, Math.Exp(Math.PI));
+                double tempMod = Math.Exp(-Phase);
+                return tempMod * LogStateMatrix.GetFrom(x)
+                    .CayleyPositive()
+                    .ConjugateOn(spectrum)
+                    .Rayleigh(StateMatrix.GetFrom(x).ColumnAverageBivariant);
+            }
+        );
+        HeatConductivity = new(x => 
+            StateMatrix.GetFrom(x).ConjugateOn(ChemistryConstant.EnergyConductivitySpectrum)
+            .Rayleigh(LogStateMatrix.GetFrom(x) * StateMatrix.GetFrom(x).ColumnAverageBivariant)
+        );
+        ResetAllCache();
+    }
+
+    /// <summary>
+    /// singleton used to represent no mixture
+    /// </summary>
+    public static readonly HomogeneousMixture Null = new();
+
+    private IEnumerable<BaseBond> ConnectedIndex()
+    {
+        HashSet<BaseBond> bs = new();
+        foreach (Compound c in Compounds)
+        {
+            if(c.Amount.IsEqualApprox(0))
+                continue;
+
+
+            foreach (BaseBond b in c.Bonds)
+            {
+                if (b.Compound.Amount.IsEqualApprox(0))
+                    throw new Exception("P??");
+
+                if (b.Compound != c)
+                    throw new Exception("??!!?");
+                if (
+                    b.Connected &&
+                    b.Energy > b.AntiBondingEnergy &&
+                    b.ConnectedBond.Energy > b.ConnectedBond.AntiBondingEnergy &&
+                    !bs.Contains(b.ConnectedBond)
+                )
+                {
+                    bs.Add(b);
+                }
+            }
+        }
+
+        return bs;
+        //Compounds.Where(c => !(0.5 * c.Amount).IsEqualApprox(0)).SelectMany(x => x.ConnectedIndex());
+    }
+    public void RPC()
+    {
+
+        int s1 = 0;
+        int s2 = 0;
+        int s3 = 0;
+        int s4 = 0;
+        foreach (Compound c in Compounds)
+        {
+            foreach (BaseBond b in c.Bonds)
+            {
+                switch (b.BondingGroup)
+                {
+                    case 0:
+                        s1++;
+                        break;
+                    case 1:
+                        s2++;
+                        break;
+                    case 2:
+                        s3++;
+                        break;
+                    default:
+                        s4++;
+                        break;
+                }
+            }
+        }
+        Console.WriteLine($"{s1} ----  {s2} ---- {s3} --- {s4}");
+    }
+
+    private BondGroups ReactionGroup()
+    {
+        HashSet<BaseBond> group1 = new();
+        HashSet<BaseBond> group2 = new();
+        HashSet<BaseBond> group3 = new();
+        HashSet<BaseBond> group4 = new();
+        foreach (Compound c in Compounds)
+        {
+            if ((c.Amount * 0.125).IsEqualApprox(0) || c.Atoms.Count > 15) 
+                continue;
+            if(c.Expression.Equals("P2So"))
+                Console.WriteLine("?");
+            foreach(BaseBond b in c.Bonds)
+            {
+                if(b.Connected || !(b.Energy > b.BondingEnergy))
+                    continue;
+                switch (b.BondingGroup)
+                {
+                    case 0:
+                        group1.Add(b);
+                        break;
+                    case 1:
+                        group2.Add(b);
+                        break;
+                    case 2:
+                        group3.Add(b);
+                        break;
+                    case 3:
+                        group4.Add(b);
+                        break;
+                }
+            }
+        }
+        return new(group1, group2, group3, group4);
+    }
+
+    /// <summary>
+    /// elasticity range: 0, 4pi
+    /// elasticity controls the speed k drops from e^pi to e^-4pi
+    /// elasticity -> 0; k -> constant
+    /// b -> 4pi; k -> e^pi
+    /// b -> -4pi; k -> e^-4pi
+    /// </summary>
+    public CacheItem<double> CompressionElasticity { get; }
+
+    /// <summary>
+    /// bias range: -4pi, 4pi
+    /// bias controls the temp of triple point
+    /// higher b -> higher melting point and higher boiling point
+    /// </summary>
+    public CacheItem<double> CompressionBias { get; }
+
+    /// <summary>
+    /// phase -> 1 implies material is more like gas
+    /// phase -> -1 implies material is more like solid
+    /// 
+    /// </summary>
+    public double Phase => Compounds.Sum(c => c.Concentration * c.Phase);
+
+    /// <summary>
+    /// stiffness range: e^-4pi, e^pi
+    /// higher stiffness -> harder to compress -> solid like
+    /// lower stiffness -> easier to compress -> gas like
+    /// Stiffness K, Density D, Free Density D0, Temperature T
+    /// P = K(1/D - 1/D0)
+    /// 
+    /// Temperature range: -pi, pi
+    /// </summary>
+    public double CompressionStiffness => Math.Exp(Math.PI / 2d * (5d * Phase - 3d));
+
+    /// <summary>
+    /// e^-4pi ... e^pi
+    /// </summary>
+    public CacheItem<double> FreeDensity { get; }
+
+    /// <summary>
+    /// volume that under 0 compression
+    /// </summary>
+    public double FreeVolume => Amount / FreeDensity.Get;
+
+    /// <summary>
+    /// P = K (1/D - 1/D0)
+    /// </summary>
+    public double Pressure => (Volume - FreeVolume) * CompressionStiffness;
+
+    /// <summary>
+    /// Volume of the mixture
+    /// </summary>
+    public double Volume { get; set; }
+
+    /// <summary>
+    /// Heat transfer ability
+    /// </summary>
+    public CacheItem<double> HeatConductivity { get; }
+
+    /// <summary>
+    /// energy / amount = temperature
+    /// </summary>
+    public double Temperature
+    {
+        get => Energy / Amount;
+        set => Energy = value * Amount;
+    }
+    /// <summary>
+    /// total Energy the mixture holds 
+    /// </summary>
+    public double Energy
+    {
+        get => Compounds.Sum(c => c.Energy);
+        set
+        {
+            foreach (Compound c in Compounds)
+                c.Energy = value * c.Concentration;
+        }
+    }
+    /// <summary>
+    /// reset all cache
+    /// </summary>
+    public void ResetAllCache()
+    {
+        LayerCache = LinkNode<HomogeneousMixture>.Null;
+    }
+    /// <summary>
+    /// the heterogeneous mixture that holds this homogeneous mixture
+    /// </summary>
+    public HeterogeneousMixture HeterogeneousMixture { get; set; } = HeterogeneousMixture.Null;
+
+    /// <summary>
+    /// Compounds of this mixture
+    /// </summary>
+    public HashSet<Compound> Compounds { get; set; } = new HashSet<Compound>();
+    /// <summary>
+    /// Compounds whose ratio is greater than 1%
+    /// </summary>
+    public IEnumerable<Compound> MainCompounds => Compounds.Where(c => c.Concentration > 0.001d);
+    private LinkNode<HomogeneousMixture> LayerCache { get; set; } = LinkNode<HomogeneousMixture>.Null;
+    /// <summary>
+    /// 
+    /// </summary>
+    public LinkNode<HomogeneousMixture> Layer
+    {
+        get => LayerCache == LinkNode<HomogeneousMixture>.Null
+            ? HeterogeneousMixture.LayerOrder.Find(this)
+            : LayerCache;
+        set => LayerCache = value;
+    }
+
+    /// <summary>
+    /// State matrix of this mixture
+    /// </summary>
+    public CacheItem<SU3> StateMatrix { get; }
+
+    /// <summary>
+    /// Log of the state matrix
+    /// </summary>
+    public CacheItem<LieSU3> LogStateMatrix { get; }
+    
+    /// <summary>
+    /// Split the homogeneous mixture into another heterogeneous mixture by ratio or amount;
+    /// </summary>
+    /// <param name="amount"> The amount or ratio</param>
+    /// <param name="into">another heterogeneous mixture</param>
+    /// <param name="byRatio">determine if the amount variable is ratio or amount</param>
+    /// <returns></returns>
+    public HomogeneousMixture Split(double amount, HeterogeneousMixture into, bool byRatio = false)
+    {
+        double ebs = Energy;
+        double vbs = Volume;
+        double ratio = byRatio ? amount : amount / Amount;
+        HomogeneousMixture res = new HomogeneousMixture();
+        foreach (Compound c in Compounds)
+        {
+            Compound split = c.Split(amount, byRatio);
+            if(split != Compound.Null)
+                res.AddCompound(split);
+        }
+        res.Energy = ebs*ratio;
+        res.Volume = vbs*ratio;
+        Energy = ebs * (1 - ratio);
+        Volume = vbs * (1 - ratio);
+        into.AddLayer(res);
+        return res;
+    }
+
+    /// <summary>
+    /// Add compound into the homogeneous mixture
+    /// </summary>
+    /// <param name="compound">compound to add</param>
+    /// <param name="resetCache"></param>
+    /// <param name="skipCombine">add without combine to isomorphic compound</param>
+    public void AddCompound(Compound compound, bool resetCache = true, bool skipCombine=false)
+    {
+        if (compound.HomogeneousMixture == this)
+            return;
+        bool added = false;
+        if(!skipCombine)
+        {
+            foreach (Compound sCompound in Compounds.Where(x => x.Expression == compound.Expression))
+            {
+                if (sCompound.IsometricTo(compound))
+                {
+                    sCompound.Amount += compound.Amount;
+                    compound.Amount = 0d;
+                    sCompound.Energy += compound.Energy;
+                    added = true;
+                    break;
+                }
+            }
+        }
+
+        compound.HomogeneousMixture = added ? Null : this;
+        if (!added)
+            Compounds.Add(compound);
+        if(resetCache)
+            ResetAllCache();
+    }
+    /// <summary>
+    /// Amount of this mixture 
+    /// </summary>
+    public double Amount
+    {
+        get => Compounds
+            .Select(c => c.Amount)
+            .DefaultIfEmpty(0)
+            .Sum();
+        set
+        {
+            Dictionary<Compound, double> res = new Dictionary<Compound, double>();
+            foreach (Compound c in Compounds)
+                res[c] = value * c.Concentration;
+            foreach (Compound c in Compounds)
+                c.Amount = res[c];
+        }
+    }
+
+    /// <summary>
+    /// ratio of this mixture in the heterogeneous mixture
+    /// </summary>
+    public double Ratio =>
+        Amount / (HeterogeneousMixture == HeterogeneousMixture.Null ? Amount : HeterogeneousMixture.Amount);
+
+    
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="homo"></param>
+    public void Resolve(HomogeneousMixture homo)
+    {
+        foreach (Compound c in homo.Compounds)
+            Resolve(c, false);
+        ResetAllCache();
+        
+    }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="compound"></param>
+    /// <param name="resetCache"></param>
+    public void Resolve(Compound compound, bool resetCache = true)
+    {
+        AddCompound(compound, resetCache);
+    }
+    
+    /// <summary>
+    /// density of the mixture, amount of molecular per volume
+    /// </summary>
+    public double Density => Amount / Volume;
+    
+    /// <summary>
+    /// save mixture into string
+    /// </summary>
+    /// <returns></returns>
+    public string Dump() =>
+        $"`ENV?ENG{Energy.ExactDoubleString()}?V{Volume.ExactDoubleString()}`%ENV`COMPS{String.Join(HomogeneousMixtureResolver.CompoundsSplit, Compounds.Select(c => c.Dump()))}`%COMPS";
+
+    /// <summary>
+    /// update volume
+    /// </summary>
+    public void VolumeUpdate()
+    {
+        double sVolume = Volume + HeterogeneousMixture.ForceFromContainer / CompressionStiffness;
+        Volume = Math.Max(sVolume, 1E-6);
+    }
+
+    /// <summary>
+    /// exchange energy to neighbor mixtures 
+    /// </summary>
+    public void HeatExchange()
+    {
+        if (Layer != Layer.Parent.First)
+        {
+            HomogeneousMixture mPrevious = Layer.Previous.Value;
+            double balanceTemp = (Energy + mPrevious.Energy) / (Amount + mPrevious.Amount);
+            double dTemp = balanceTemp - Temperature;
+            double conductivity = Math.Sqrt(HeatConductivity.Get * mPrevious.HeatConductivity.Get);
+            double transferTemp = dTemp * conductivity;
+            Temperature += transferTemp;
+            mPrevious.Temperature -= transferTemp;
+        }
+        if (Layer != Layer.Parent.Last)
+        {
+            HomogeneousMixture mNext = Layer.Next.Value;
+            double balanceTemp = (Energy + mNext.Energy) / (Amount + mNext.Amount);
+            double dTemp = balanceTemp - Temperature;
+            double conductivity = Math.Sqrt(HeatConductivity.Get * mNext.HeatConductivity.Get);
+            double transferTemp = dTemp * conductivity;
+            Temperature += transferTemp;
+            mNext.Temperature -= transferTemp;
+        }
+
+        double envBalanceTemp = HeterogeneousMixture.Container.EnvironmentTemperature;
+        double envDTemp = envBalanceTemp - Temperature;
+        double envConductivity = HeatConductivity.Get;
+        Temperature += envDTemp * envConductivity;
+    }
+    private void ReactConnecting(HashSet<BaseBond> g1, HashSet<BaseBond> g2)
+    {
+        double maxCd = -1;
+        BaseBond xb1 = BaseBond.Null;
+        BaseBond xb2 = BaseBond.Null;
+        foreach (BaseBond b1 in g1)
+        {
+            foreach (BaseBond b2 in g2)
+            {
+                H3 cd = new(Complex.ImaginaryOne / 2 * (b1.LogState.Get + b2.LogState.Get));
+                double cdX = cd.Rayleigh((b1.LogState.Get + b2.LogState.Get).ColumnAverage);
+                if (cdX > maxCd)
+                {
+                    maxCd = cdX;
+                    xb1 = b1;
+                    xb2 = b2;
+                }
+            }
+        }
+        if (xb1 == BaseBond.Null)
+            return;
+        if (xb1 == xb2)
+        {
+            PlainPack2<Compound, BaseBond> s = xb1.Compound.SplitWithBondMap(0.5, xb2, true);
+            xb2 = s.Item2;
+        }
+
+        if (xb1.Compound.Amount.IsSignificantlyGreaterThen(xb2.Compound.Amount))
+        {
+            double amountDiff = xb1.Compound.Amount - xb2.Compound.Amount;
+            Compound oth = xb1.Compound.Split(amountDiff);
+            xb1.HomogeneousMixture.AddCompound(oth);
+        }
+
+        if (xb1.Compound.Amount.IsSignificantlyLessThen(xb2.Compound.Amount))
+        {
+            double amountDiff = xb2.Compound.Amount - xb1.Compound.Amount;
+            Compound oth = xb2.Compound.Split(amountDiff);
+            xb2.HomogeneousMixture.AddCompound(oth);
+        }
+
+        string log = $"From Composite React of {xb1.Compound.Amount}*{xb1.Compound.Expression} and {xb2.Compound.Amount}*{xb2.Compound.Expression}";
+        
+        xb1.Connect(xb2);
+        xb1.Compound.TraceLog.Add(log);
+    }
+
+    private int ReactionType()
+    {
+        Complex d = LogStateMatrix.Get.CayleyPositive().Det();
+        return d.Phase switch
+        {
+            > 7 * Math.PI / 4 or < Math.PI / 4 => 0,
+            > Math.PI / 4 and < 3 * Math.PI / 4 => 1,
+            > 3 * Math.PI / 4 and < 5 * Math.PI / 4 => 2,
+            _ => 3
+        };
+    }
+
+    /// <summary>
+    /// react within the homogeneous mixture
+    /// </summary>
+    public void React()
+    {
+        BaseBond[] connected = ConnectedIndex().ToArray();
+        foreach (BaseBond bond in connected)
+        {
+            if (bond.Compound.Amount.IsEqualApprox(0))
+                continue;
+            if (
+                bond.Energy > bond.AntiBondingEnergy &&
+                bond.ConnectedBond.Energy > bond.ConnectedBond.AntiBondingEnergy
+            )
+            {
+                bond.Disconnect();
+            }
+            
+        }
+        for (int i = 1; i <= 3; i++)
+        {
+            BondGroups bg = ReactionGroup();
+            int type = ReactionType();
+            if(type == 0)
+            {
+                ReactConnecting(bg.Group1, bg.Group3);
+                ReactConnecting(bg.Group2, bg.Group4);
+            }
+            else if (type == 1)
+            {
+                ReactConnecting(bg.Group1, bg.Group1);
+                ReactConnecting(bg.Group3, bg.Group3);
+            }
+            else if (type == 2)
+            {
+                ReactConnecting(bg.Group2, bg.Group2);
+                ReactConnecting(bg.Group3, bg.Group3);
+            }
+            else
+            {
+                ReactConnecting(bg.Group1, bg.Group4);
+                ReactConnecting(bg.Group2, bg.Group3);
+            }
+        }
+
+    }
+    /// <summary>
+    /// higher level mixture can resolve lower level mixture
+    /// </summary>
+    public double ResolveLevel => Compounds.Sum(c => c.Concentration * c.ResolveLevel);
+    /// <summary>
+    /// ability to resolve other compound
+    /// </summary>
+    /// <param name="c"></param>
+    /// <returns></returns>
+    public double Resolvability(Compound c)
+    {
+        LieSU3 s = c.LogStateMatrix.Get ^ LogStateMatrix.Get;
+        C3 a = new(1, 0.5, 1);
+        C3 b = new(0.5, 1, 0.5);
+        Complex w = a * s * b;
+        w = ComplexFieldStructure.Structure.Fix(w);
+        double res = -Math.Tan((Math.Sin(w.Phase) * Math.PI - Math.PI) / 2);
+        if (Math.Abs(res) < 1E-6)
+            return 0;
+        return res;
+    }
+    /// <summary>
+    ///  a tensor of order 3 abstracted from this mixture
+    /// </summary>
+    public C3 Ita => Compounds
+        .Select(c => c.Concentration * c.StateMatrix.Get.ColumnAverageBivariant)
+        .Aggregate((a, b) => a + b);
+    /// <summary>
+    /// Red component  
+    /// </summary>
+    public Byte ColorRed => (Byte)
+    (
+        StateMatrix.Get
+            .ConjugateOn(ChemistryConstant.OpticalFilter.RedDefinite)
+            .Rayleigh(Ita) * 255
+    );
+    /// <summary>
+    /// Green component
+    /// </summary>
+    public Byte ColorGreen => (Byte)
+    (
+        StateMatrix.Get
+            .ConjugateOn(ChemistryConstant.OpticalFilter.GreenDefinite)
+            .Rayleigh(Ita) * 255
+    );
+    /// <summary>
+    /// Blue component
+    /// </summary>
+    public Byte ColorBlue => (Byte)
+    (
+        StateMatrix.Get
+            .ConjugateOn(ChemistryConstant.OpticalFilter.BlueDefinite)
+            .Rayleigh(Ita) * 255
+    );
+    /// <summary>
+    /// Transparent component
+    /// </summary>
+    public Byte ColorTransparent => (Byte)
+    (
+        StateMatrix.Get
+            .ConjugateOn(ChemistryConstant.OpticalFilter.OpacityDefinite)
+            .Rayleigh(Ita) * 255
+    );
+
+    /// <summary>
+    /// Should not be used by any methods except ResolveNext and ResolvePrev
+    /// </summary>
+    private void PureResolve(HomogeneousMixture hOther)
+    {
+        HashSet<Compound> toAbsorb = new ();
+        foreach (Compound c in hOther.Compounds)
+        {
+            double maxRes = Resolvability(c);
+            double oRes = hOther.Resolvability(c);
+            if(oRes < 0 || maxRes < 0 || maxRes < oRes)
+                continue;
+            double maxResAmount = Amount;
+            foreach (Compound ct in Compounds)
+                if (ct.IsometricTo(c))
+                    maxResAmount -= ct.Amount;
+            maxResAmount *= maxRes;
+            if(maxResAmount < 0)
+                continue;
+            double splitAmount = Math.Min(maxResAmount, c.Amount);
+            if(splitAmount.IsEqualApprox(0))
+                continue;
+            string log = $"{splitAmount} split from {c.Amount} * {c.Expression}";
+            Compound cSplit = c.Split(splitAmount);
+            cSplit.TraceLog.Add(log);
+            toAbsorb.Add(cSplit);
+        }
+        foreach (Compound c in toAbsorb)
+            AddCompound(c);
+        if(hOther.Amount.IsEqualApprox(0))
+            HeterogeneousMixture.RemoveLayer(hOther);
+    }
+    private void ResolveNext()
+    {
+        if (HeterogeneousMixture == HeterogeneousMixture.Null || Layer.Next.IsTail)
+            return;
+        PureResolve(Layer.Next.Value);
+    }
+    private void ResolvePrevious()
+    {
+        if (HeterogeneousMixture == HeterogeneousMixture.Null || Layer.Previous.IsHead)
+            return;
+        PureResolve(Layer.Previous.Value);
+    }
+    
+    private bool Resolve()
+    {
+        ResolveNext();
+        ResolvePrevious();
+        return false;
+    }
+
+    private bool Precipitate()
+    {
+        foreach (Compound c in Compounds)
+        {
+            double maxRes = Resolvability(c);
+            if (maxRes < 0)
+                continue;
+            double maxResAmount = maxRes * (Amount - c.Amount);
+            if(maxResAmount.IsEqualApprox(0))
+                continue;
+
+            double aDiff = c.Amount - maxResAmount; 
+            if (aDiff.IsSignificantlyGreaterThen(0) )
+            {
+                Compound oth = c.Split(c.Amount - maxResAmount);
+                if (oth.FreeDensity.Get > FreeDensity.Get)
+                {
+                    if (Layer.Next.IsEnding)
+                    {
+                        HomogeneousMixture m = HeterogeneousMixture.AddLayer();
+                        m.Layer.MoveAfter(Layer);
+                        m.AddCompound(oth);
+                        m.Volume = m.FreeVolume;
+                        return true;
+                    }
+                    Layer.Next.Value.AddCompound(oth);
+                }
+                else
+                {
+                    if (Layer.Previous.IsEnding)
+                    {
+                        HomogeneousMixture m = HeterogeneousMixture.AddLayer();
+                        m.Layer.MoveBefore(Layer);
+                        m.AddCompound(oth);
+                        m.Volume = m.FreeVolume;
+                        return true;
+                    }
+                    Layer.Previous.Value.AddCompound(oth);
+                }
+            }
+        }
+        return false;
+    }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <returns>if any new layer created or deleted</returns>
+    public bool PrecipitateAndResolve()
+    {
+        bool a = Resolve();
+        bool b = Precipitate();
+        return a || b;
+    }
+
+    /// <summary>
+    /// combine identical compounds in the miture
+    /// </summary>
+    public void Degenerate()
+    {
+        HashSet<Compound> toRemove = new();
+        foreach (IGrouping<string, Compound> g in Compounds.GroupBy(c => c.Expression))
+        {
+            HashSet<Compound> classified = new();
+            foreach (Compound c in g)
+            {
+                classified.Add(c);
+                foreach (Compound cx in g.Where(x => !classified.Contains(x)))
+                {
+                    if (!c.IsometricTo(cx))
+                        continue;
+                    classified.Add(cx);
+                    c.Amount += cx.Amount;
+                    cx.Amount = 0;
+                    toRemove.Add(cx);
+                }
+            }
+        }
+
+        foreach (Compound c in toRemove)
+        {
+            c.HomogeneousMixture = Null;
+            Compounds.Remove(c);
+        }
+    }
+
+    public static void Test()
+    {
+        Func<double, double> a = x => -Math.Tan((Math.PI * x - Math.PI) / 2);
+        Console.WriteLine(a(0));
+        Console.WriteLine(a(-1));
+        Console.WriteLine(a(1));
+    }
+}

src/Chemistry/Mixtures/Resolver/HeterogeneousMixtureResolver.cs

@@ -0,0 +1,32 @@
+using System.Text.RegularExpressions;
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+
+namespace VirtualChemistry.Chemistry.Mixtures.Resolver;
+
+/// <summary>
+/// get heterogeneous mixture from descriptions
+/// </summary>
+public static class HeterogeneousMixtureResolver
+{
+    private const string LayersRegex = @"`LAYERS(.*)`%LAYERS";
+    private static readonly Regex Matcher = new Regex(LayersRegex);
+    /// <summary>
+    /// split different homogeneous mixtures
+    /// </summary>
+    public const string LayerSplit = "<LayerSp>";
+    
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="dumpString"></param>
+    /// <returns></returns>
+    public static HeterogeneousMixture Resolve(string dumpString)
+    {
+        HeterogeneousMixture res = new HeterogeneousMixture();
+        string layers = Matcher.Match(dumpString).Result("$1");
+        foreach (string layerString in layers.Split(LayerSplit))
+            res.AddLayer(HomogeneousMixtureResolver.Resolve(layerString), false);
+        return res;
+    }
+
+}

src/Chemistry/Mixtures/Resolver/HomogeneousMixtureResolver.cs

@@ -0,0 +1,45 @@
+using System.Text.RegularExpressions;
+using Skeleton.Utils;
+using VirtualChemistry.Chemistry.Compounds.Resolver;
+using VirtualChemistry.Chemistry.Mixtures.Implements;
+
+namespace VirtualChemistry.Chemistry.Mixtures.Resolver;
+
+/// <summary>
+/// get homogeneous mixture from description
+/// </summary>
+public static class HomogeneousMixtureResolver
+{
+    private const string EnvironmentRegex = @"`ENV(.*)`%ENV";
+    private const string CompoundsRegex = @"`COMPS(.*)`%COMPS";
+    private const string MatchString = EnvironmentRegex + CompoundsRegex;
+    private static readonly Regex Matcher = new Regex(MatchString);
+    
+    private const string VolumeRegex = @"\?V\(([0-9,]+)\)";
+    private const string EnergyRegex = @"\?E\(([0-9,]+)\)";
+    private const string EnvMatchString = VolumeRegex + EnergyRegex;
+    private static readonly Regex EnvMatcher = new Regex(EnvMatchString);
+    /// <summary>
+    /// split different compounds
+    /// </summary>
+    public const string CompoundsSplit = "<CompSp>";
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="dumpString"></param>
+    /// <returns></returns>
+    public static HomogeneousMixture Resolve(string dumpString)
+    {
+        Match result = Matcher.Match(dumpString);
+        HomogeneousMixture res = new HomogeneousMixture();
+        string envString = result.Result("$1");
+        Match envRes = EnvMatcher.Match(envString);
+        double volume = envRes.Result("$1").ByteStringToDouble();
+        double energy = envRes.Result("$2").ByteStringToDouble();
+        foreach (string compoundString in result.Result("$2").Split(CompoundsSplit))
+            res.AddCompound(CompoundResolver.Resolve(compoundString));
+        res.Volume = volume;
+        res.Energy = energy;
+        return res;
+    }
+}

src/Constants/CachedKeys.cs

@@ -0,0 +1,103 @@
+namespace VirtualChemistry.Constants;
+
+/// <summary>
+/// keys for tensors
+/// </summary>
+public static class CachedKeys
+{
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string StateMatrix = $"{TypePrefix.SU3}_StateMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string LogStateMatrix = $"{TypePrefix.LieSU3}_LogStateMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string StructureMatrix = $"{TypePrefix.SU3}_StructureMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string LogConnectionMatrix = $"{TypePrefix.LieSU3}_LogConnectionMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string EnergyDistributionFactor = $"{TypePrefix.Complex}_EnergyDistributionFactor";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string HeatConductivity = $"{TypePrefix.Double}_HeatConductivity";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string FreeDensity = $"{TypePrefix.Double}_FreeDensity";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string CompressionStiffness = $"{TypePrefix.Double}_CompressionStiffness";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string CompressionElasticity = $"{TypePrefix.Double}_CompressionElasticity";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string CompressionBias = $"{TypePrefix.Double}_CompressionBias";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string BondInformationMatrix = $"{TypePrefix.SU3}_BondInformationMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string LogEigenStateMatrix = $"{TypePrefix.LieSU3}_LogEigenStateMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string LogKernelMatrix = $"{TypePrefix.LieSU3}_LogKernelMatrix";
+    /// <summary>
+    /// 
+    /// </summary>
+    public const string HalfBondInformation = $"{TypePrefix.LieSU3}_HalfBondInformation";
+}
+/// <summary>
+/// tensor type info
+/// </summary>
+public static class TypePrefix
+{
+    /// <summary>
+    /// real
+    /// </summary>
+    public const string Double = "D";
+    /// <summary>
+    /// complex
+    /// </summary>
+    public const string Complex = "C";
+    /// <summary>
+    /// c3
+    /// </summary>
+    public const string C3 = "C3";
+    /// <summary>
+    /// c33
+    /// </summary>
+    public const string C33 = "C33";
+    /// <summary>
+    /// u3
+    /// </summary>
+    public const string U3 = "U3";
+    /// <summary>
+    /// su3
+    /// </summary>
+    public const string SU3 = "SU3";
+    /// <summary>
+    /// lie u3
+    /// </summary>
+    public const string LieU3 = "LieU3";
+    /// <summary>
+    /// lie su3
+    /// </summary>
+    public const string LieSU3 = "LieSU3";
+    
+}

src/Constants/ChemistryConstant.cs

@@ -0,0 +1,653 @@
+using System.Numerics;
+using Skeleton.Constants;
+using VirtualChemistry.Utils.Helpers;
+
+namespace VirtualChemistry.Constants;
+
+/// <summary>
+/// chemistry related constants
+/// </summary>
+public static class ChemistryConstant
+{
+    private static readonly Complex I = Complex.ImaginaryOne;
+    /// <summary>
+    /// volume of a single atom
+    /// </summary>
+    public static readonly double AtomVolume = Math.Exp(-2 * Math.PI);
+
+    /// <summary>
+    /// to calculate edv for atom
+    /// </summary>
+    public static readonly C3 AtomEnergyDistributionVector = new C3
+    (
+        Complex.Exp(Math.PI * I / 4d),
+        Complex.Exp(Math.PI * I * 11d / 12d),
+        Complex.Exp(Math.PI * I * 19d / 12d)
+    );
+    /// <summary>
+    /// to calculate edv for bond
+    /// </summary>
+    public static readonly C3 BondEnergyDistributionVector = new C3
+    (
+        Complex.Exp(Math.PI * I * 5d / 4d),
+        Complex.Exp(Math.PI * 23d / 12d),
+        Complex.Exp(Math.PI * I * 7d / 12d)
+    );
+
+    /// <summary>
+    /// to calculate edv from atom
+    /// </summary>
+    public static readonly DiagR3 AtomEnergyDistributionSpectrum = new DiagR3(1d / 4d, 11d / 12d, 19d / 12d);
+    /// <summary>
+    /// to calculate edv for bond
+    /// </summary>
+    public static readonly DiagR3 BondEnergyDistributionSpectrum = new DiagR3(5d / 4d, 23d / 12d, 7d / 12d);
+    /// <summary>
+    /// to calculate energy conductivity
+    /// </summary>
+    public static readonly DiagR3 EnergyConductivitySpectrum = new DiagR3(2d/15d, 7d/15d, 11d/15d);
+    /// <summary>
+    /// all bond types
+    /// </summary>
+    public static class BondTypes
+    {
+        /// <summary>
+        /// type m
+        /// </summary>
+        public const string M = "m";
+        /// <summary>
+        /// type s
+        /// </summary>
+        public const string S = "s";
+        /// <summary>
+        /// type d
+        /// </summary>
+        public const string D = "d";
+        /// <summary>
+        /// type y
+        /// </summary>
+        public const string Y = "y";
+        /// <summary>
+        /// type q
+        /// </summary>
+        public const string Q = "q";
+        /// <summary>
+        /// type p
+        /// </summary>
+        public const string P = "p";
+        /// <summary>
+        /// type h
+        /// </summary>
+        public const string H = "h";
+        /// <summary>
+        /// all types
+        /// </summary>
+        public static readonly string[] All = new[] { M, S, D, Y, Q, P, H };
+    }
+
+    /// <summary>
+    /// all atom types
+    /// </summary>
+    public static class Elements
+    {
+        /// <summary>
+        /// atom q
+        /// </summary>
+        public const string Q = "Q";
+        /// <summary>
+        /// atom r
+        /// </summary>
+        public const string R = "R";
+        /// <summary>
+        /// atom es
+        /// </summary>
+        public const string Es = "Es";
+        /// <summary>
+        /// atom d
+        /// </summary>
+        public const string D = "D";
+        /// <summary>
+        /// atom ue
+        /// </summary>
+        public const string Ue = "Ue";
+        /// <summary>
+        /// atom m
+        /// </summary>
+        public const string M = "M";
+        /// <summary>
+        /// atom k
+        /// </summary>
+        public const string K = "K";
+        /// <summary>
+        /// atom p
+        /// </summary>
+        public const string P = "P";
+        /// <summary>
+        /// atom so
+        /// </summary>
+        public const string So = "So";
+        /// <summary>
+        /// atom e
+        /// </summary>
+        public const string E = "E";
+        /// <summary>
+        /// atom u
+        /// </summary>
+        public const string U = "U";
+        /// <summary>
+        /// atom a
+        /// </summary>
+        public const string A = "A";
+        /// <summary>
+        /// atom cx
+        /// </summary>
+        public const string Cx = "Cx";
+        /// <summary>
+        /// atom v
+        /// </summary>
+        public const string V = "V";
+        /// <summary>
+        /// atom vi
+        /// </summary>
+        public const string Vi = "Vi";
+    }
+
+    private static double AtomicMass(int atomicNumber) =>
+        Math.Tan(ChemistryHelper.AtomHeightOrder(atomicNumber) * Math.PI / 2d);
+
+    private static SU3 AtomEigenState(int atomicNumber)
+    {
+        SU3 dam = ChemistryHelper.AtomAdjointStateMatrix(atomicNumber);
+        return dam.ConjugateOn(ChemistryHelper.AtomKernel(atomicNumber));
+    }
+
+    /// <summary>
+    /// adjoint matrices for atom
+    /// </summary>
+    public static class AtomAdjoints
+    {
+        /// <summary>
+        /// for q
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateQ = ChemistryHelper.AtomAdjointStateMatrix(1);
+        /// <summary>
+        /// for r
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateR = ChemistryHelper.AtomAdjointStateMatrix(2);
+        /// <summary>
+        /// for es
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateEs = ChemistryHelper.AtomAdjointStateMatrix(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateD = ChemistryHelper.AtomAdjointStateMatrix(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateUe = ChemistryHelper.AtomAdjointStateMatrix(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateM = ChemistryHelper.AtomAdjointStateMatrix(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateK = ChemistryHelper.AtomAdjointStateMatrix(7);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateP = ChemistryHelper.AtomAdjointStateMatrix(8);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateSo = ChemistryHelper.AtomAdjointStateMatrix(9);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateE = ChemistryHelper.AtomAdjointStateMatrix(10);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateU = ChemistryHelper.AtomAdjointStateMatrix(11);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateA = ChemistryHelper.AtomAdjointStateMatrix(12);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateCx = ChemistryHelper.AtomAdjointStateMatrix(13);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateV = ChemistryHelper.AtomAdjointStateMatrix(14);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomAdjointStateVi = ChemistryHelper.AtomAdjointStateMatrix(15);
+    }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static class AtomKernels
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateQ = ChemistryHelper.AtomKernel(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateR = ChemistryHelper.AtomKernel(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateEs = ChemistryHelper.AtomKernel(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateD = ChemistryHelper.AtomKernel(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateUe = ChemistryHelper.AtomKernel(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateM = ChemistryHelper.AtomKernel(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateK = ChemistryHelper.AtomKernel(7);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateP = ChemistryHelper.AtomKernel(8);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateSo = ChemistryHelper.AtomKernel(9);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateE = ChemistryHelper.AtomKernel(10);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateU = ChemistryHelper.AtomKernel(11);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateA = ChemistryHelper.AtomKernel(12);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateCx = ChemistryHelper.AtomKernel(13);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateV = ChemistryHelper.AtomKernel(14);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomKernelStateVi = ChemistryHelper.AtomKernel(15);
+
+    }
+
+    /// <summary>
+    /// atom eigen state matrices
+    /// </summary>
+    public static class AtomEigenStates
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateQ = AtomEigenState(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateR = AtomEigenState(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateEs = AtomEigenState(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateD = AtomEigenState(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateUe = AtomEigenState(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateM = AtomEigenState(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateK = AtomEigenState(7);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateP = AtomEigenState(8);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateSo = AtomEigenState(9);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateE = AtomEigenState(10);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateU = AtomEigenState(11);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateA = AtomEigenState(12);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateCx = AtomEigenState(13);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateV = AtomEigenState(14);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 AtomEigenStateVi = AtomEigenState(15);
+    }
+
+    /// <summary>
+    /// atomic masses for atoms 
+    /// </summary>
+    public static class AtomicMasses
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassQ = AtomicMass(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassR = AtomicMass(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassEs = AtomicMass(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassD = AtomicMass(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassUe = AtomicMass(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassM = AtomicMass(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassK = AtomicMass(7);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassP = AtomicMass(8);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassSo = AtomicMass(9);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassE = AtomicMass(10);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassU = AtomicMass(11);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassA = AtomicMass(12);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassCx = AtomicMass(13);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassV = AtomicMass(14);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly double AtomicMassVi = AtomicMass(15);
+        
+    }
+    
+    private static SU3 BondEigenState(int bondNumber)
+    {
+        SU3 dam = ChemistryHelper.BondAdjointStateMatrix(bondNumber);
+        return dam.ConjugateOn(ChemistryHelper.BondKernel(bondNumber));
+    }
+    /// <summary>
+    /// kernel matrices for bonds
+    /// </summary>
+    public static class BondKernels
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelM = ChemistryHelper.BondKernel(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelS = ChemistryHelper.BondKernel(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelD = ChemistryHelper.BondKernel(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelY = ChemistryHelper.BondKernel(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelQ = ChemistryHelper.BondKernel(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelP = ChemistryHelper.BondKernel(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondKernelH = ChemistryHelper.BondKernel(7);
+    }
+    /// <summary>
+    /// adjoint matrices for bond
+    /// </summary>
+    public static class BondAdjoints
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointM = ChemistryHelper.BondAdjointStateMatrix(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointS = ChemistryHelper.BondAdjointStateMatrix(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointD = ChemistryHelper.BondAdjointStateMatrix(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointY = ChemistryHelper.BondAdjointStateMatrix(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointQ = ChemistryHelper.BondAdjointStateMatrix(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointP = ChemistryHelper.BondAdjointStateMatrix(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondAdjointH = ChemistryHelper.BondAdjointStateMatrix(7);
+    }
+
+    /// <summary>
+    /// eigen state matrices for bond
+    /// </summary>
+    public static class BondEigenStates
+    {
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateM = BondEigenState(1);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateS = BondEigenState(2);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateD = BondEigenState(3);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateY = BondEigenState(4);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateQ = BondEigenState(5);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateP = BondEigenState(6);
+        /// <summary>
+        /// 
+        /// </summary>
+        public static readonly SU3 BondEigenStateH = BondEigenState(7);
+        
+    }
+
+    /// <summary>
+    /// max bond numbers for specific bond number
+    /// </summary>
+    /// <param name="bondNumber"></param>
+    /// <returns></returns>
+    public static int BondMaxNumbers(int bondNumber) => bondNumber switch
+    {
+        1 => 1,
+        2 => 2,
+        3 => 1,
+        4 => 2,
+        5 => 2,
+        6 => 1,
+        7 => 2,
+        _ => -1
+    };
+
+    /// <summary>
+    /// filters to calculate the color of mixtures
+    /// </summary>
+    public static class OpticalFilter
+    {
+        /// <summary>
+        /// to calculate redness
+        /// </summary>
+        public static readonly U3 RedFilter = new LieU3
+        (
+            I,0, -1,
+            0, I, 0,
+            1, 0, -I
+        ).Exp();
+
+        /// <summary>
+        /// to calculate greenness
+        /// </summary>
+        public static readonly U3 GreenFilter = new LieU3
+        (
+            0, -I, 0,
+            -I, 0, -1,
+            0, 1, 0
+        ).Exp();
+        /// <summary>
+        /// to calculate blueness
+        /// </summary>
+        public static readonly U3 BlueFilter = new LieU3
+        (
+            0, 0, -I,
+            0, 0, -1,
+            -I, 1, 0
+        ).Exp();
+        /// <summary>
+        /// to calculate alpha channel 
+        /// </summary>
+        public static readonly U3 OpacityFilter = new LieU3(
+            I, -I, -I,
+            -I, I, -1,
+            -I, 1, -I
+        ).Exp();
+
+        /// <summary>
+        /// red
+        /// </summary>
+        public static readonly H3 RedDefinite = RedFilter.ConjugateOn(AlgebraConstant.UniformSpectrum3);
+        /// <summary>
+        /// green
+        /// </summary>
+        public static readonly H3 GreenDefinite = GreenFilter.ConjugateOn(AlgebraConstant.UniformSpectrum3);
+        /// <summary>
+        /// blue
+        /// </summary>
+        public static readonly H3 BlueDefinite = BlueFilter.ConjugateOn(AlgebraConstant.UniformSpectrum3);
+        /// <summary>
+        /// alpha
+        /// </summary>
+        public static readonly H3 OpacityDefinite = OpacityFilter.ConjugateOn(new DiagR3(0.05d, 0.74d, 1d));
+    }
+
+
+    /// <summary>
+    /// all elements
+    /// </summary>
+    public static readonly string[] ElementSymbols =
+    {
+        "Q", "R", "Es", "D", "Ue",
+        "M", "K", "P", "So", "E",
+        "U", "A", "Cx", "V", "Vi"
+    };
+    
+    
+    /// <summary>
+    /// x
+    /// </summary>
+    public static class CommonResources
+    {
+        /// <summary>
+        /// x
+        /// </summary>
+        public const string Letroline = "res://Data/Chemicals/HighEnergyMixtureLetroline.btx";
+    }
+
+    /// <summary>
+    /// Used to measure Cayley value of homogeneous mixture
+    /// </summary>
+    public static readonly C3 CayleyMeasure = new (1 - I, I,  -1 + I);
+    /// <summary>
+    /// Used to measure Euclid value of homogeneous mixture
+    /// </summary>
+    public static readonly C3 EuclidMeasure = new(1, I,  -I);
+    
+}

src/DataStructure/Caches/Cache.cs

@@ -0,0 +1,54 @@
+using Skeleton.Utils.Helpers;
+
+namespace VirtualChemistry.DataStructure.Caches;
+
+/// <summary>
+/// cache to store a specific type of data
+/// </summary>
+/// <typeparam name="TValue"></typeparam>
+public class Cache<TValue> : Dictionary<string, TValue>, ICache
+{
+    /// <inheritdoc />
+    public Cache()
+    {
+        UpdateRequest = new HashSet<string>();
+    }
+
+    /// <summary>
+    /// indexer
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter">re calculate by getter if key is expired of no value stored in the specified key</param>
+    public TValue this[string key, Func<TValue> getter]
+    {
+        get
+        {
+            if(ValueOutdated(key))
+            {
+                UpdateRequest.Remove(key);
+                return this[key] = getter();
+            }
+            return this[key];
+        }
+    }
+
+    /// <inheritdoc />
+    public void DeleteCache()
+    {
+        Keys.ForEach(x => Remove(x));
+        UpdateRequest = new HashSet<string>();
+    }
+
+    /// <inheritdoc />
+    public void DeleteCache(string key) => Remove(key);
+    private HashSet<string> UpdateRequest { get; set; }
+    private bool ValueOutdated(string key) => !ContainsKey(key) || UpdateRequest.Contains(key);
+
+    /// <inheritdoc />
+    public void Expire(string key) => UpdateRequest.Add(key);
+
+    /// <inheritdoc />
+    public void Expire(IEnumerable<string> keys) => keys.ForEach(Expire);
+
+
+}

src/DataStructure/Caches/GeneralCache.cs

@@ -0,0 +1,156 @@
+using System.Numerics;
+using Skeleton.Utils.Helpers;
+
+namespace VirtualChemistry.DataStructure.Caches;
+
+/// <summary>
+/// a general propose cache to store many type of tensors
+/// </summary>
+public class GeneralCache
+{
+    private Cache<C3> C3Cache { get; set; } = new();
+    private Cache<C33> C33Cache { get; set; } = new();
+    private Cache<Complex> ComplexCache { get; set; } = new();
+    private Cache<double> DoubleCache { get; set; } = new();
+    private Cache<U3> U3Cache { get; set; } = new();
+    private Cache<SU3> SU3Cache { get; set; } = new();
+    private Cache<LieU3> LieU3Cache { get; set; } = new();
+    private Cache<LieSU3> LieSU3Cache { get; set; } = new();
+
+    private static T Identity<T>(T x) => x;
+    private Func<T, T> RegisterOnExpire<T>(string key, Action? e)
+    {
+        
+        if (e != null && !OnExpire.ContainsKey(key))
+            OnExpire[key] = e;
+        return Identity<T>;
+    }
+
+    /// <summary>
+    /// C3 tensor getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire">a function to expire other keys if this key is expired</param>
+    public C3 this[string key, Func<C3> getter, Action? onExpire = null] =>
+        RegisterOnExpire<C3>(key, onExpire)(C3Cache[key, getter]);
+
+    /// <summary>
+    /// C33 tensor getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public C33 this[string key, Func<C33> getter, Action? onExpire = null] =>
+        RegisterOnExpire<C33>(key, onExpire)(C33Cache[key, getter]);
+    /// <summary>
+    /// complex getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public Complex this[string key, Func<Complex> getter, Action? onExpire = null] =>
+        RegisterOnExpire<Complex>(key, onExpire)(ComplexCache[key, getter]);
+    /// <summary>
+    /// real getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public double this[string key, Func<double> getter, Action? onExpire = null] =>
+        RegisterOnExpire<double>(key, onExpire)(DoubleCache[key, getter]);
+
+    /// <summary>
+    /// U3 getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public U3 this[string key, Func<U3> getter, Action? onExpire = null] =>
+        RegisterOnExpire<U3>(key, onExpire)(U3Cache[key, getter]);
+
+    /// <summary>
+    /// SU3 getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public SU3 this[string key, Func<SU3> getter, Action? onExpire = null] =>
+        RegisterOnExpire<SU3>(key, onExpire)(SU3Cache[key, getter]);
+
+    /// <summary>
+    /// LieU3 getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public LieU3 this[string key, Func<LieU3> getter, Action? onExpire = null] =>
+        RegisterOnExpire<LieU3>(key, onExpire)(LieU3Cache[key, getter]);
+
+    /// <summary>
+    /// LieSU3 getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="getter"></param>
+    /// <param name="onExpire"></param>
+    public LieSU3 this[string key, Func<LieSU3> getter, Action? onExpire = null] =>
+        RegisterOnExpire<LieSU3>(key, onExpire)(LieSU3Cache[key, getter]);
+
+    /// <summary>
+    /// general propose getter
+    /// </summary>
+    /// <param name="key"></param>
+    /// <exception cref="Exception"></exception>
+    public ICache this[string key] => key.Split('_')[0] switch
+    {
+        "D" => DoubleCache,
+        "C" => ComplexCache,
+        "C3" => C3Cache,
+        "C33" => C33Cache,
+        "U3" => U3Cache,
+        "SU3" => SU3Cache,
+        "LieU3" => LieU3Cache,
+        "LieSU3" => LieSU3Cache,
+        _ => throw new Exception()
+    };
+    
+    /// <summary>
+    /// general expire
+    /// </summary>
+    /// <param name="key"></param>
+    public void Expire(string key)
+    {
+        if (OnExpire.ContainsKey(key))
+            OnExpire[key]();
+        this[key].Expire(key);
+    }
+
+    /// <summary>
+    /// general expire
+    /// </summary>
+    /// <param name="keys"></param>
+    public void Expire(IEnumerable<string> keys) => keys.ForEach(Expire);
+    /// <summary>
+    /// general delete cache
+    /// </summary>
+    /// <param name="key"></param>
+    public void DeleteCache(string key) => this[key].DeleteCache(key);
+
+    /// <summary>
+    /// trigger the action when key is expired
+    /// </summary>
+    public Dictionary<string, Action> OnExpire { get; set; } = new();
+
+    /// <summary>
+    /// get raw real(constant)
+    /// </summary>
+    /// <param name="key"></param>
+    /// <returns></returns>
+    public double GetRaw(string key) => DoubleCache.GetValueOrDefault(key);
+    /// <summary>
+    /// init a constant real
+    /// </summary>
+    /// <param name="key"></param>
+    /// <param name="value"></param>
+    public void SetRaw(string key, double value) => DoubleCache[key] = value;
+}

src/DataStructure/Caches/ICache.cs

@@ -0,0 +1,27 @@
+namespace VirtualChemistry.DataStructure.Caches;
+
+/// <summary>
+/// interface for cache 
+/// </summary>
+public interface ICache
+{
+    /// <summary>
+    /// delete all cache
+    /// </summary>
+    public void DeleteCache();
+    /// <summary>
+    /// delete for a specific key
+    /// </summary>
+    /// <param name="key"></param>
+    public void DeleteCache(string key);
+    /// <summary>
+    /// expire a key, the value need to be re calculated from getter
+    /// </summary>
+    /// <param name="key"></param>
+    public void Expire(string key);
+    /// <summary>
+    /// expire a set of keys
+    /// </summary>
+    /// <param name="keys"></param>
+    public void Expire(IEnumerable<string> keys);
+}

src/DataStructure/Packs/BondGroups.cs

@@ -0,0 +1,33 @@
+using Skeleton.DataStructure.Packs;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+
+namespace VirtualChemistry.DataStructure.Packs;
+
+using BondGroup = HashSet<BaseBond>;
+
+/// <inheritdoc />
+public class BondGroups : Pack4<BondGroup, BondGroup, BondGroup, BondGroup>
+{
+    /// <summary>
+    /// 
+    /// </summary>
+    public BondGroup Group1 => Item1;
+    /// <summary>
+    /// 
+    /// </summary>
+    public BondGroup Group2 => Item2;
+    /// <summary>
+    /// 
+    /// </summary>
+    public BondGroup Group3 => Item3;
+    /// <summary>
+    /// 
+    /// </summary>
+    public BondGroup Group4 => Item4;
+
+    /// <inheritdoc />
+    public BondGroups(BondGroup g1, BondGroup g2, BondGroup g3, BondGroup g4) : base(g1, g2, g3, g4)
+    {
+        
+    }
+}

src/DataStructure/Packs/CompoundMap.cs

@@ -0,0 +1,67 @@
+using Skeleton.DataStructure.Packs;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+using VirtualChemistry.Chemistry.Compounds.Implements;
+
+namespace VirtualChemistry.DataStructure.Packs;
+
+/// <summary>
+/// compound dump map
+/// </summary>
+public class CompoundDumpMap : Pack3<Dictionary<BaseAtom, string>, Dictionary<BaseBond, string>, string>
+{
+    /// <summary>
+    /// atom map
+    /// </summary>
+    public Dictionary<BaseAtom, string> AtomMap => Item1;
+    /// <summary>
+    /// bond map
+    /// </summary>
+    public Dictionary<BaseBond, string> BondMap => Item2;
+    /// <summary>
+    /// save string
+    /// </summary>
+    public string DumpString => Item3;
+
+    /// <summary>
+    /// constructor
+    /// </summary>
+    /// <param name="atomMap"></param>
+    /// <param name="bondMap"></param>
+    /// <param name="dumpString"></param>
+    public CompoundDumpMap(Dictionary<BaseAtom, string> atomMap, Dictionary<BaseBond, string> bondMap, string dumpString) :
+        base(atomMap, bondMap, dumpString)
+    {
+    }
+}
+
+/// <summary>
+/// resolve map for compound
+/// </summary>
+public class CompoundResolveMap : Pack3<Dictionary<string, BaseAtom>, Dictionary<string, BaseBond>, Compound>
+{
+    /// <summary>
+    /// atom map
+    /// </summary>
+    public Dictionary<string, BaseAtom> AtomMap => Item1;
+    /// <summary>
+    /// bond map
+    /// </summary>
+    public Dictionary<string, BaseBond> BondMap => Item2;
+    /// <summary>
+    /// created compound
+    /// </summary>
+    public Compound ResolvedCompound => Item3;
+
+    /// <summary>
+    /// constructor
+    /// </summary>
+    /// <param name="atomMap"></param>
+    /// <param name="bondMap"></param>
+    /// <param name="resolvedCompound"></param>
+    public CompoundResolveMap(Dictionary<string, BaseAtom> atomMap, Dictionary<string, BaseBond> bondMap,
+        Compound resolvedCompound) :
+        base(atomMap, bondMap, resolvedCompound)
+    {
+    }
+}

src/DataStructure/Packs/ConnectionInfo.cs

@@ -0,0 +1,37 @@
+using Skeleton.DataStructure.Packs;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+
+namespace VirtualChemistry.DataStructure.Packs;
+
+/// <summary>
+/// connection info of an atom and a bond
+/// </summary>
+public class ConnectionInfo : Pack3<BaseBond, BaseBond, BaseAtom>
+{
+    /// <summary>
+    /// constructor
+    /// </summary>
+    /// <param name="bondFrom"></param>
+    /// <param name="bondTo"></param>
+    /// <param name="connectedAtom"></param>
+    public ConnectionInfo(BaseBond bondFrom, BaseBond bondTo, BaseAtom connectedAtom)
+    {
+        Item1 = bondFrom;
+        Item2 = bondTo;
+        Item3 = connectedAtom;
+    }
+
+    /// <summary>
+    /// from bond
+    /// </summary>
+    public BaseBond BondFrom => Item1;
+    /// <summary>
+    /// to bond
+    /// </summary>
+    public BaseBond BondTo => Item2;
+    /// <summary>
+    /// to atom
+    /// </summary>
+    public BaseAtom ConnectedAtom => Item3;
+}

src/DataStructure/Packs/FullConnectionInfo.cs

@@ -0,0 +1,64 @@
+using Skeleton.DataStructure.Packs;
+using VirtualChemistry.Chemistry.Atoms.Implements;
+using VirtualChemistry.Chemistry.Bonds.Implements;
+
+namespace VirtualChemistry.DataStructure.Packs;
+
+/// <summary>
+/// all connections from one atom to another
+/// </summary>
+public class FullConnectionInfo : Pack3<BaseAtom, HashSet<BaseBond>, BaseAtom>
+{
+    /// <summary>
+    /// constructor
+    /// </summary>
+    /// <param name="bond"></param>
+    public FullConnectionInfo(BaseBond bond) : base(bond.Atom, new HashSet<BaseBond>() { bond }, bond.ConnectedBond.Atom)
+    {
+    }
+
+    /// <summary>
+    /// from atom
+    /// </summary>
+    public BaseAtom Atom1 => Item1;
+    /// <summary>
+    /// connected bonds
+    /// </summary>
+    public HashSet<BaseBond> Bonds => Item2;
+    /// <summary>
+    /// bonds of atom 1
+    /// </summary>
+    public BaseAtom Atom2 => Item3;
+    private string OnwRepresentation => $"{Atom1.Element}-[{String.Join('|', Bonds.Select(bond => $"{bond.BondType}-{bond.ConnectedBond.BondType}"))}]-{Atom2.Element}";
+    private string RevRepresentation => $"{Atom2.Element}-[{String.Join('|', Bonds.Select(bond => $"{bond.ConnectedBond.BondType}-{bond.BondType}"))}]-{Atom1.Element}";
+
+    /// <summary>
+    /// save string
+    /// </summary>
+    public string Representation => Atom1.AtomicNumber + Bonds.Sum(bond => bond.BondNumber) >
+                                    Atom2.AtomicNumber + Bonds.Sum(bond => bond.ConnectedBond.BondNumber)
+        ? OnwRepresentation
+        : RevRepresentation;
+
+    /// <summary>
+    /// try to add a bond to bonds
+    /// </summary>
+    /// <param name="bond"></param>
+    /// <returns></returns>
+    public bool TryAbsorbBond(BaseBond bond)
+    {
+        if(bond.Atom == Atom1 && bond.ConnectedBond.Atom == Atom2)
+        {
+            Bonds.Add(bond);
+            return true;
+        }
+
+        if (bond.Atom == Atom2 && bond.ConnectedBond.Atom == Atom1)
+        {
+            Bonds.Add(bond.ConnectedBond);
+            return true;
+        }
+
+        return false;
+    }
+}

src/Using.cs

@@ -0,0 +1,42 @@
+global using C2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<System.Numerics.Complex>.FVector;
+global using C3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<System.Numerics.Complex>.FVector;
+global using C4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<System.Numerics.Complex>.FVector;
+
+global using R2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<double>.FVector;
+global using R3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<double>.FVector;
+global using R4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<double>.FVector;
+
+global using C22 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<System.Numerics.Complex>.FMatrix;
+global using C33 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<System.Numerics.Complex>.FMatrix;
+global using C44 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<System.Numerics.Complex>.FMatrix;
+
+global using R22 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<double>.FMatrix;
+global using R33 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<double>.FMatrix;
+global using R44 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<double>.FMatrix;
+
+global using U2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.FUnitaryMatrix;
+global using U3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.FUnitaryMatrix;
+global using U4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.FUnitaryMatrix;
+
+global using SU2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.FSpecialUnitaryMatrix;
+global using SU3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.FSpecialUnitaryMatrix;
+global using SU4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.FSpecialUnitaryMatrix;
+
+global using LieU2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.FLieUnitaryMatrix;
+global using LieU3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.FLieUnitaryMatrix;
+global using LieU4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.FLieUnitaryMatrix;
+
+global using LieSU2 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.FSpecialLieUnitaryMatrix;
+global using LieSU3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.FSpecialLieUnitaryMatrix;
+global using LieSU4 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.FSpecialLieUnitaryMatrix;
+
+global using C2Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<System.Numerics.Complex>.FVectorSpace;
+global using C3Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<System.Numerics.Complex>.FVectorSpace;
+global using C4Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<System.Numerics.Complex>.FVectorSpace;
+
+global using R2Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim2>.OnField<double>.FVectorSpace;
+global using R3Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<double>.FVectorSpace;
+global using R4Space = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim4>.OnField<double>.FVectorSpace;
+
+global using DiagR3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.OnField<double>.FDiagonalMatrix;
+global using H3 = Skeleton.Algebra.CategoryOf<Skeleton.Algebra.DimensionProviders.IDim3>.FHermitianMatrix;

src/Utils/Helpers/ChemistryHelper.cs

@@ -0,0 +1,282 @@
+using System.Numerics;
+
+namespace VirtualChemistry.Utils.Helpers;
+
+/// <summary>
+/// helper functions for chemistry
+/// </summary>
+public static class ChemistryHelper
+{
+    private static readonly Complex I = Complex.ImaginaryOne;
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static double BondHeightOrder(int n) => Math.Pow(4d / 5d, n) - 1;
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static double BondWeightOrder(int n)
+    {
+        double ra = n / 11d;
+        int det = (int)Math.Floor(ra / 0.5d);
+        if (det % 2 == 0)
+            return ra % 0.5d;
+        return 0.5 - (ra % 0.5d);
+    }
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex BondHeight(int n) => Complex.Exp(I * Math.PI * BondHeightOrder(n));
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex BondWeight(int n) => Complex.Exp(I * Math.PI * BondWeightOrder(n));
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex BondE1(int n) => BondHeight(n);
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex BondE2(int n) => BondWeight(n);
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex BondE3(int n) => 1 / (BondE1(n)*BondE2(n));
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="p"></param>
+    /// <returns></returns>
+    public static SU3 BondKernel(int p) => new
+    (
+        BondE1(p), 0, 0, 
+        0, BondE2(p), 0, 
+        0, 0, BondE3(p)
+    );
+    
+    
+    
+    private static readonly double Sqrt13 = Math.Sqrt(13d);
+    private static readonly double Sqrt7 = Math.Sqrt(7d);
+    private static readonly double Sqrt3 = Math.Sqrt(3d);
+    private static readonly double Sqrt2 = Math.Sqrt(2d);
+    private static readonly double Sqrt21 = Sqrt3 * Sqrt7;
+
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly Complex BondAdjointEigen1 = I * Math.Sqrt(0.5d * (5d + Sqrt21));
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly Complex BondAdjointEigen2 = I * Math.Sqrt(0.5d * (5d - Sqrt21));
+
+    
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 BondAdjointEigenVector1 = new C3
+    (
+        1d/Sqrt2,
+        -I/2d,
+        1/2
+    );
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 BondAdjointEigenVector2 = new C3
+    (
+        0,
+        I/Sqrt2,
+        1d/Sqrt2
+    );
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 BondAdjointEigenVector3 = new C3
+    (
+        -1d/Sqrt2,
+        I/Sqrt2,
+        1d/2d
+    );
+
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C33 BondAdjointCoMatrix =
+        new C33(new[] { BondAdjointEigenVector1, BondAdjointEigenVector2, BondAdjointEigenVector3 }, false);
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="bondNumber"></param>
+    /// <returns></returns>
+    public static SU3 BondAdjointStateMatrix(int bondNumber)
+    {
+        double t = Math.Cos(bondNumber * Math.E) + Sqrt3;
+        Complex a1 = Complex.Exp(t * BondAdjointEigen1);
+        Complex a2 = Complex.Exp(t * BondAdjointEigen2);
+        Complex a3 = 1 / (a1 * a2);
+        C33 expBondAdjointEigenDiagonal = new
+        (
+            a1, 0d, 0d,
+            0d, a2, 0d,
+            0d, 0d, a3
+        );
+        return new SU3(BondAdjointCoMatrix * expBondAdjointEigenDiagonal * BondAdjointCoMatrix.Inv());
+    }
+
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static double AtomHeightOrder(int n) => 1 - Math.Pow(6d / 7d, n);
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static double AtomWeightOrder(int n)
+    {
+        double ra = n / 7d;
+
+        if ((int)Math.Floor(ra / 0.5) % 2 == 0)
+            return ra % 0.5d;
+        return 0.5d - (ra % 0.5d);
+    }
+    
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex AtomHeight(int n) => Complex.Exp(I * Math.PI * AtomHeightOrder(n));
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex AtomWeight(int n) => Complex.Exp(I * Math.PI * AtomWeightOrder(n));
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex AtomE1(int n) => AtomHeight(n);
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex AtomE2(int n) => AtomWeight(n);
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="n"></param>
+    /// <returns></returns>
+    public static Complex AtomE3(int n) => 1 / (AtomE1(n) * AtomE2(n));
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="atomicNumber"></param>
+    /// <returns></returns>
+    public static SU3 AtomKernel(int atomicNumber) => new
+    (
+        AtomE1(atomicNumber), 0, 0, 
+        0, AtomE2(atomicNumber), 0, 
+        0, 0, AtomE3(atomicNumber)
+    );
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly Complex AtomAdjointEigen1 = I * Complex.Sqrt(0.5 * (5 + Sqrt21));
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly Complex AtomAdjointEigen2 = I * Complex.Sqrt(0.5 * (5 - Sqrt21));
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 AtomAdjointEigenVector1 = new C3
+    (
+        0d,
+        -I * (Sqrt2 - 1d) / (Complex.Sqrt(4d - 2d * Sqrt2)),
+        1d / (Complex.Sqrt(4 - 2 * Sqrt2))
+    );
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 AtomAdjointEigenVector2 = new C3
+    (
+        1d,
+        0d,
+        0d
+    );
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly C3 AtomAdjointEigenVector3 = new C3
+    (
+        0d,
+        I * (1d + Sqrt2) / Complex.Sqrt(2d * (2d + Sqrt2)),
+        1d / Complex.Sqrt(2d * (2d + Sqrt2))
+    );
+
+
+    /// <summary>
+    /// 
+    /// </summary>
+    public static readonly SU3 AtomAdjointCoMatrix =
+        new ( new []{ AtomAdjointEigenVector1, AtomAdjointEigenVector2, AtomAdjointEigenVector3 }, false);
+
+    /// <summary>
+    /// 
+    /// </summary>
+    /// <param name="atomicNumber"></param>
+    /// <returns></returns>
+    public static SU3 AtomAdjointStateMatrix(int atomicNumber)
+    {
+        double t = Math.Cos(atomicNumber * Math.E) + Sqrt3;
+        Complex a1 = Complex.Exp(t * AtomAdjointEigen1);
+        Complex a2 = Complex.Exp(t * AtomAdjointEigen2);
+        Complex a3 = 1 / (a1 * a2);
+        SU3 expAtomAdjointEigenDiagonal = new SU3
+        (
+            a1, 0, 0, 
+            0, a2 , 0, 
+            0, 0, a3 
+        );
+        return AtomAdjointCoMatrix.ConjugateOn(expAtomAdjointEigenDiagonal);
+    }
+
+    
+}