/// <summary>
/// Creates all fermion terms generated by all symmetries of an orbital integral.
/// </summary>
/// <param name="nOrbitals">Total number of distinct orbitals.</param>
/// <param name="orbitalIntegral">Input orbital integral.</param>
/// <param name="indexConvention">Indexing scheme from spin-orbitals to integers.</param>
/// <returns>List of fermion terms generated by all symmetries of an orbital integral.</returns>
public static IEnumerable <(HermitianFermionTerm, double)> ToHermitianFermionTerms(
this OrbitalIntegral orbitalIntegral,
int nOrbitals,
IndexConvention indexConvention = IndexConvention.UpDown)
{
var termType = orbitalIntegral.TermType;
if (termType == TermType.OrbitalIntegral.OneBody)
{
return(orbitalIntegral.ToOneBodySpinOrbitalTerms(nOrbitals, indexConvention));
}
else if (termType == TermType.OrbitalIntegral.TwoBody)
{
return(orbitalIntegral.ToTwoBodySpinOrbitalTerms(nOrbitals, indexConvention));
}
else if (termType == TermType.OrbitalIntegral.Identity)
{
return(new List <(HermitianFermionTerm, double)>()
{
(new HermitianFermionTerm(), orbitalIntegral.Coefficient)
});
}
else
{
throw new System.NotImplementedException();
}
}
/// <summary>
/// Creates all fermion terms generated by all symmetries of a one-body orbital integral.
/// </summary>
/// <param name="nOrbitals">Total number of distinct orbitals.</param>
/// <param name="orbitalIntegral">Input orbital integral.</param>
/// <param name="indexConvention">Indexing scheme from spin-orbitals to integers.</param>
private static IEnumerable <(HermitianFermionTerm, double)> ToOneBodySpinOrbitalTerms(
this OrbitalIntegral orbitalIntegral,
int nOrbitals,
IndexConvention indexConvention)
{
// One-electron orbital integral symmetries
// ij = ji
var pqSpinOrbitals = orbitalIntegral.EnumerateOrbitalSymmetries().EnumerateSpinOrbitals();
var coefficient = orbitalIntegral.Coefficient;
foreach (var pq in pqSpinOrbitals)
{
var pInt = pq[0].ToInt(indexConvention, nOrbitals);
var qInt = pq[1].ToInt(indexConvention, nOrbitals);
if (pInt == qInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt }.ToLadderSequence()), orbitalIntegral.Coefficient);
}
else if (pInt < qInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt }.ToLadderSequence()), 2.0 * orbitalIntegral.Coefficient);
}
}
}
/// <summary>
/// Updates an instance of <see cref="FermionHamiltonian"/>
/// with all spin-orbitals from described by a sequence of four-body orbital integrals.
/// </summary>
/// <param name="nOrbitals">Total number of distinct orbitals.</param>
/// <param name="orbitalIntegral">Sequence of four-body orbital integrals.</param>
/// <param name="indexConvention">Indexing scheme from spin-orbitals to integers.</param>
private static IEnumerable <(HermitianFermionTerm, double)> ToTwoBodySpinOrbitalTerms(
this OrbitalIntegral orbitalIntegral,
int nOrbitals,
IndexConvention indexConvention)
{
List <(HermitianFermionTerm, double)> fermionTerms = new List <(HermitianFermionTerm, double)>();
// Two-electron orbital integral symmetries
// ijkl = lkji = jilk = klij = ikjl = ljki = kilj = jlik.
var pqrsSpinOrbitals = orbitalIntegral.EnumerateOrbitalSymmetries().EnumerateSpinOrbitals();
var coefficient = orbitalIntegral.Coefficient;
// We only need to see one of these.
// Now iterate over pqrsArray
foreach (var pqrs in pqrsSpinOrbitals)
{
var p = pqrs[0];
var q = pqrs[1];
var r = pqrs[2];
var s = pqrs[3];
var pInt = p.ToInt(indexConvention, nOrbitals);
var qInt = q.ToInt(indexConvention, nOrbitals);
var rInt = r.ToInt(indexConvention, nOrbitals);
var sInt = s.ToInt(indexConvention, nOrbitals);
// Only consider terms on the lower diagonal due to Hermitian symmetry.
// For terms with two different orbital indices, possibilities are
// PPQQ (QQ = 0), PQPQ, QPPQ (p<q), PQQP, QPQP (p<q), QQPP (PP=0)
// Hence, if we only count PQQP, and PQPQ, we need to double the coefficient.
// iU jU jU iU | iU jD jD iD | iD jU jU iD | iD jD jD iD
if (pInt == sInt && qInt == rInt && pInt < qInt)
{ // PQQP
yield return(new HermitianFermionTerm(new[] { pInt, qInt, rInt, sInt }.ToLadderSequence()), 1.0 * coefficient);
}
else if (pInt == rInt && qInt == sInt && pInt < qInt)
{
// iU jU iU jU | iD jD iD jD
// PQPQ
yield return(new HermitianFermionTerm(new[] { pInt, qInt, sInt, rInt }.ToLadderSequence()), -1.0 * coefficient);
}
else if (qInt == rInt && pInt < sInt && rInt != sInt && pInt != qInt)
{
// PQQR
// For any distinct pqr, [i;j;j;k] generates PQQR ~ RQQP ~ QPRQ ~ QRPQ. We only need to record one.
if (rInt < sInt)
{
if (pInt < qInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, sInt, rInt }.ToLadderSequence()), -2.0 * coefficient);
}
else
{
yield return(new HermitianFermionTerm(new[] { qInt, pInt, sInt, rInt }.ToLadderSequence()), 2.0 * coefficient);
}
}
else
{
if (pInt < qInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, rInt, sInt }.ToLadderSequence()), 2.0 * coefficient);
}
else
{
yield return(new HermitianFermionTerm(new[] { qInt, pInt, rInt, sInt }.ToLadderSequence()), -2.0 * coefficient);
}
}
}
else if (qInt == sInt && pInt < rInt && rInt != sInt && pInt != sInt)
{
// PQRQ
// For any distinct pqr, [i;j;k;j] generates {p, q, r, q}, {q, r, q, p}, {q, p, q, r}, {r, q, p, q}. We only need to record one.
if (pInt < qInt)
{
if (rInt > qInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, rInt, sInt }.ToLadderSequence()), 2.0 * coefficient);
}
else
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, sInt, rInt }.ToLadderSequence()), -2.0 * coefficient);
}
}
else
{
yield return(new HermitianFermionTerm(new[] { qInt, pInt, rInt, sInt }.ToLadderSequence()), -2.0 * coefficient);
}
}
else if (pInt < qInt && pInt < rInt && pInt < sInt && qInt != rInt && qInt != sInt && rInt != sInt)
{
// PQRS
// For any distinct pqrs, [i;j;k;l] generates
// {{p, q, r, s}<->{s, r, q, p}<->{q, p, s, r}<->{r, s, p, q},
// {1,2,3,4}<->{4,3,2,1}<->{2,1,4,3}<->{3,4,1,2}
// {p, r, q, s}<->{s, q, r, p}<->{r, p, s, q}<->{q, s, p, r}}
// 1324, 4231, 3142, 2413
if (rInt < sInt)
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, sInt, rInt }.ToLadderSequence()), -2.0 * coefficient);
}
else
{
yield return(new HermitianFermionTerm(new[] { pInt, qInt, rInt, sInt }.ToLadderSequence()), 2.0 * coefficient);
}
}
}
}