# N-body matrix elements

The matrix element of an $N$-body operator between two Slater determinants may be expanded according to the Löwdin rules (which reduce to the Slater–Condon rules if all single-particle orbitals are orthogonal):

$$$$$\label{eqn:matrix-element-expansion} \matrixel{\Phi_A}{\Omega_n}{\Phi_B} = \frac{1}{n!}\sum_p (-)^p \matrixel{k_1k_2...k_n}{\Omega_n}{l_1l_2...l_n} D^{AB}({k_1k_2...k_n}|{l_1l_2...l_n})$$$$$

where $D^{AB}({k_1k_2...k_n}|{l_1l_2...l_n})$ is the determinant minor of the orbital overlap determinant $D^{AB}$ with the rows ${k_1k_2...k_n}$ and columns ${l_1l_2...l_n}$ stricken out, and $p$ runs over all permutations.

In general, a term in the expansion is thus of the form

$$$$$\alpha\matrixel{k_1k_2...k_n}{\Omega_n}{l_1l_2...l_n}\braket{a}{b}\braket{c}{d}\dots\braket{y}{z},$$$$$

where $\alpha$ is a scalar. This is represented by NBodyTerm type.

EnergyExpressions.OrbitalOverlapType
OrbitalOverlap(a,b)

Represents the overlap between the orbitals a and b in a N-body matrix element expansion.

Examples

julia> EnergyExpressions.OrbitalOverlap(:a,:b)
⟨a|b⟩
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EnergyExpressions.OrbitalMatrixElementType
OrbitalMatrixElement(a,o,b)

Represents the N-body matrix element between the sets of orbitals a and b.

Examples

julia> struct MyTwoBodyOperator <: TwoBodyOperator end

julia> EnergyExpressions.OrbitalMatrixElement((:a,:b), MyTwoBodyOperator(), (:c,:d))
⟨a b|MyTwoBodyOperator()|c d⟩
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EnergyExpressions.numbodiesFunction
numbodies(::NBodyOperator{N})

Returns the number of bodies coupled by the N-body operator, i.e. N.

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numbodies(lco::LinearCombinationOperator)

Returns the maximum number of bodies coupled by any of the N-body operators in the LinearCombinationOperator.

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numbodies(::OrbitalOverlap)

Returns the number of bodies coupled by the zero-body operator in the orbital overlap, i.e. 0.

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numbodies(::OrbitalMatrixElement{N})

Returns the number of bodies coupled by the operator, i.e. N.

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EnergyExpressions.NBodyMatrixElementMethod
NBodyMatrixElement(a, op, b, overlap)

Generate the matrix element of op, a linear combination of NBodyOperator, between the configurations (e.g. Slater determinants) a and b, according to the Löwdin rules. The matrix overlap contains the mutual overlaps between all single-particle orbitals in the Slater determinants. If the orbitals are all orthogonal, the Löwdin rules collapse to the Slater–Condon rules.

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EnergyExpressions.NBodyMatrixElementMethod
NBodyMatrixElement(a, op, b, nzcofactors)

Generate the matrix element of the N-body operator op, between the orbital sets a and b, where nzcofactors list all N-tuples for which the determinantal cofactor of the orbital overlap matrix is non-vanishing.

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EnergyExpressions.NBodyMatrixElementMethod
NBodyMatrixElement(a, op, b, overlap)

Generate the matrix element of the N-body operator op, between the Slater determinants a and b, according to the Löwdin rules. The matrix overlap contains the mutual overlaps between all single-particle orbitals in the Slater determinants. If the orbitals are all orthogonal, the Löwdin rules collapse to the Slater–Condon rules.

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EnergyExpressions.isdependentFunction
isdependent(o::OrbitalOverlap, orbital)

Returns true if the OrbitalOverlap o depends on orbital.

Examples

julia> isdependent(OrbitalOverlap(:a,:b), :a)
false

julia> isdependent(OrbitalOverlap(:a,:b), Conjugate(:a))
true

julia> isdependent(OrbitalOverlap(:a,:b), :b)
true
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isdependent(o::OrbitalMatrixElement, orbital)

Returns true if the OrbitalMatrixElement o depends on orbital.

Examples

julia> isdependent(EnergyExpressions.OrbitalMatrixElement((:a,), OneBodyHamiltonian(), (:b,)), :a)
false

julia> isdependent(EnergyExpressions.OrbitalMatrixElement((:a,), OneBodyHamiltonian(), (:b,)), Conjugate(:a))
true

julia> isdependent(EnergyExpressions.OrbitalMatrixElement((:a,), OneBodyHamiltonian(), (:b,)), :b)
true

julia> isdependent(EnergyExpressions.OrbitalMatrixElement((:a,:b,), CoulombInteraction(), (:c,:d)), :c)
true
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isdependent(nbt::NBodyTerm, o)

Returns true if any of the factors comprising nbt is dependent on the orbital o. Not that the result is dependent on whether o is conjugated or not.

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EnergyExpressions.transformFunction
transform(f::Function, nbt::NBodyTerm)

Transform integrals of the the N-body matrix element expansion term nbt according to the function f, which should accept a single NBodyTermFactor as its argument.

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transform(f::Function, nbme::NBodyMatrixElement)

Transform integrals of the the N-body matrix element nbme according to the function f, which should accept a single NBodyTermFactor as its argument, and return a NBodyMatrixElement. This is useful for adapting energy expressions to specific symmetries of the system under consideration.

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EnergyExpressions.overlap_matrixFunction
overlap_matrix(a::Cfg, b::Cfg[, overlaps=[]]) where Cfg

Generate the single-particle orbital overlap matrix, between the orbitals in the configurations (e.g. Slater determinants) a and b. All orbitals are assumed to be orthogonal, except for those which are given in overlaps.

Examples

First we define two Slater determinants that have some orbitals in common:

julia> sa = SlaterDeterminant([:i, :j, :l,:k̃])
i(1)j(2)l(3)k̃(4) - i(1)j(2)l(4)k̃(3) - i(1)j(3)l(2)k̃(4) + i(1)j(3)l(4)k̃(2) + …  + i(4)j(1)l(3)k̃(2) + i(4)j(2)l(1)k̃(3) - i(4)j(2)l(3)k̃(1) - i(4)j(3)l(1)k̃(2) + i(4)j(3)l(2)k̃(1)

julia> sb = SlaterDeterminant([:i, :j, :k, :l̃])
i(1)j(2)k(3)l̃(4) - i(1)j(2)k(4)l̃(3) - i(1)j(3)k(2)l̃(4) + i(1)j(3)k(4)l̃(2) + …  + i(4)j(1)k(3)l̃(2) + i(4)j(2)k(1)l̃(3) - i(4)j(2)k(3)l̃(1) - i(4)j(3)k(1)l̃(2) + i(4)j(3)k(2)l̃(1)

The orbital overlap matrix by default is

julia> overlap_matrix(sa, sb)
4×4 SparseArrays.SparseMatrixCSC{EnergyExpressions.NBodyTerm,Int64} with 2 stored entries:
[1, 1]  =  1
[2, 2]  =  1

which has only two non-zero entries, since only two of the orbitals are common between the Slater determinants sa and sb.

We can then define that the orbitals k̃ and l̃ are non-orthogonal:

julia> overlap_matrix(sa, sb, [OrbitalOverlap(:k̃,:l̃)])
4×4 SparseArrays.SparseMatrixCSC{EnergyExpressions.NBodyTerm,Int64} with 3 stored entries:
[1, 1]  =  1
[2, 2]  =  1
[4, 4]  =  ⟨k̃|l̃⟩

We can even specify that the orbital k̃ is non-orthogonal to itself (this can be useful when the k̃ is a linear combination of orthogonal orbitals):

julia> overlap_matrix(sa, sa, [OrbitalOverlap(:k̃,:k̃)])
4×4 SparseArrays.SparseMatrixCSC{EnergyExpressions.NBodyTerm,Int64} with 4 stored entries:
[1, 1]  =  1
[2, 2]  =  1
[3, 3]  =  1
[4, 4]  =  ⟨k̃|k̃⟩

Notice that this overlap matrix was calculated between the Slater determinant sa and itself.

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EnergyExpressions.compareFunction
compare(a::NBodyMatrixElement, op, b::NBodyMatrixElement; kwargs...)

Compare the NBodyMatrixElements a and b for similarity; all the terms of a need to be present in b, and vice versa, and their expansion coefficients have to agree when compared using op.

This function is mainly designed for testing purposes, i.e. to compare an expression with a reference, generated otherwise. It may not be performant. It may also fail on edge cases.

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Base.:==Method
Base.:(==)(a::NBodyMatrixElement, b::NBodyMatrixElement; kwargs...)

Test if a and b are exactly equal to each other, i.e. their terms all agree exactly, as well as the expansion coefficients. The actual comparison is performed by compare.

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Base.isapproxMethod
Base.isapprox(a::NBodyMatrixElement, b::NBodyMatrixElement; kwargs...)

Test if a and b are approximately equal to each other, i.e. their terms all agree exactly, and the expansion coefficients are approximately equal. The actual comparison is performed by compare.

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EnergyExpressions.EnergyExpressionType
EnergyExpression

An energy expression is given by an energy matrix, or interaction matrix, sandwiched between a vector of mixing coefficients: E = c'H*c, where c are the mixing coefficients and H the energy matrix.

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Base.MatrixType
Matrix(a, op::QuantumOperator, b[, overlaps])

Generate the matrix corresponding to the quantum operator op, between the configurations (e.g. SlaterDeterminants) a and b, i.e ⟨a|op|b⟩. It is possible to specify non-orthogonalities between single-particle orbitals in overlaps.

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Matrix(op::QuantumOperator, slater_determinants[, overlaps])

Generate the matrix corresponding to the quantum operator op, between the different slater_determinants. It is possible to specify non-orthogonalities between single-particle orbitals in overlaps.

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## Calculation of determinants

Actually computing the matrix element expansion $\eqref{eqn:matrix-element-expansion}$ is a combinatorial problem, that grows factorially with the amount of non-orthogonal orbital pairs. Furthermore, of the $(n!)^2$ terms generated from the expansion, only $n!$ are distinct, due to the integrals being symmetric with respect to interchange of the coordinates [hence the normalization factor $(n!)^{-1}$]. Thankfully, there are few symmetries that can be employed, to generate only the distinct permutations, as well as the fact that the overlap matrix is very sparse.

### Finding non-zero minors of the overlap determinant

The algorithm to find which minor determinants $\Gamma^{(N)}(k_1k_2...k_N|l_1l_2...l_N)$ do not vanish, and hence which $N$ orbitals $k_1k_2...k_N,l_1l_2...l_N$ the $N$-body operator should be contracted over, is described briefly below. It is devised to be optimal for orthogonal orbitals (i.e. linear complexity $\mathcal{O}(Nn)$ where $n$ is the number of orbitals), and near-optimal for a small amount of non-orthogonal orbitals.

Given:

• An $N$-body operator (implying $N$ rows and $N$ need to stricken out), and
• an $n\times n$ matrix, with coordinates of non-zero matrix elements: $I,J$ (from these vectors, vanishing rows/columns can easily be deduced $\implies$ $N_r$ row/$N_c$ column "rank", i.e. yet to be stricken out),

do

• find all $N_r$-combinations of the remaining rows,
• for each such combination, find all columns which would be affected if striking out that particular combination of rows,
• if the "support" (i.e. the only non-zero elements) of any of those columns vanishes when striking out the rows, that column must be stricken out, too. Total number of these columns is named $N_{cm}$,
• if more than $N_c$ columns must be stricken out ($N_{cm}>N_c$), that row combination is unviable,
• find all $N_c - N_{cm}$-combinations of the remaining columns,
• for each such combination of columns, find all rows which would be affected,
• if the support of any of those rows vanishes when striking out the candidate columns, and the row is not in the candidate set of rows to be stricken out, the column combination is unviable.
EnergyExpressions.nonzero_minorsFunction
nonzero_minors(N, overlap) -> (ks,ls)

Find all (distinct) minor determinants of order N of the orbital overlap matrix that do not vanish, i.e. all non-vanishing minors are guaranteed to be present, but not all of the returned minors are guaranteed to be non-zero. Vanishing minors returned arise when the overlap matrix is rank deficient, which is unlikely to happen when computing energy expressions, but must still be guarded against. This is most easily checked by actually calculating the cofactor, which is most likely desired anyway.

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### Contracting over the stricken out orbitals

We use Julia's built-in Base.Cartesian.@nloops iterators to span the space of all possible choices of orbitals for the contraction of orbitals. If two or more orbitals are the same, the matrix element is trivially zero (the “Fermi hole”). To avoid double-counting, we also only consider those indices that are above the hyper-diagonal.

EnergyExpressions.cofactorFunction
cofactor(k, l, A)

Calculate the cofactor of A, where the rows k and the columns l have been stricken out. The cofactor is calculated recursively, by expanding the minor determinants in cofactors, so this function should only be used in case it is known that the cofactor is non-zero.

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EnergyExpressions.distinct_permutationsFunction
distinct_permutations(fun::Function, ::NBodyOperator{N}, b)

Generate all distinct permutations p of b (which is expected to be of length N), and call fun(σ, b[p]) where σ=(-1)^p is the sign of the permutation p.

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LinearAlgebra.detFunction
det(A)

Calculate the determinant of the matrix A whose elements are of the NBodyTerm type, by expanding the determinant along the first column. This is an expensive operation, and should only be done with relatively sparse matrices.

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EnergyExpressions.@above_diagonal_loopMacro
above_diagonal_loop(N, itersym, imax, args...)

Generate N Cartesian loops for the iteration variables itersym_{1:N}, where itersym_N ∈ 1:imax, itersym_{N-1} ∈ itersym_N+1:imax, etc, i.e. above the hyper-diagonal of the N-dimensional hypercube with the side imax. args... is passed on to Base.Cartesian._nloops. above_diagonal_loop is nestable.

Examples

julia> @above_diagonal_loop 2 i 3 begin
println("==================================")
println("i = ", Base.Cartesian.@ntuple 2 i)
@above_diagonal_loop 2 j 3 begin
println("j = ", Base.Cartesian.@ntuple 2 j)
end
end
==================================
i = (2, 1)
j = (2, 1)
j = (3, 1)
j = (3, 2)
==================================
i = (3, 1)
j = (2, 1)
j = (3, 1)
j = (3, 2)
==================================
i = (3, 2)
j = (2, 1)
j = (3, 1)
j = (3, 2)
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EnergyExpressions.@anti_diagonal_loopMacro
anti_diagonal_loop(N, itersym, imax, args...)

Generate N Cartesian loops for the iteration variables itersym_{1:N}, where itersym_N ∈ 1:imax, itersym_{N-1} ∈ 1:imax\itersym_N, etc, i.e. no two iteration variables have the same values simultaneously. args... is passed on to Base.Cartesian._nloops; however, preexpr is already used to skip the diagonal elements. anti_diagonal_loop is nestable.

Examples

julia> @anti_diagonal_loop 3 i 3 begin
println("-----------------------------")
t = (Base.Cartesian.@ntuple 3 i)
println("$t: ", allunique(t)) @anti_diagonal_loop 2 j 2 begin u = (Base.Cartesian.@ntuple 2 j) println("$u: ", allunique(u))
end
end
-----------------------------
(3, 2, 1): true
(2, 1): true
(1, 2): true
-----------------------------
(2, 3, 1): true
(2, 1): true
(1, 2): true
-----------------------------
(3, 1, 2): true
(2, 1): true
(1, 2): true
-----------------------------
(1, 3, 2): true
(2, 1): true
(1, 2): true
-----------------------------
(2, 1, 3): true
(2, 1): true
(1, 2): true
-----------------------------
(1, 2, 3): true
(2, 1): true
(1, 2): true
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## Occupation number representation

As an alternative to SlaterDeterminant, we also provide an implementation of the occupation number representation (i.e. second quantization). Every configuration is thus represented as a bit vector, indicating the occupation of a specific orbital by a true value. Excitations between two configurations are easily computed using bitwise operations. The present implementation is inspired by

• Scemama, A., & Giner, E. (2013). An efficient implementation of Slater–Condon rules. CoRR, arXiv:1311.6244,

but extends upon it by also supporting non-orthogonal orbitals.

The benefit of this approach is much more efficient identification of which cofactors in $\eqref{eqn:matrix-element-expansion}$ are non-zero, than the approach taken in nonzero_minors.

EnergyExpressions.BitConfigurationsType
BitConfigurations(orbitals, configurations)

Represent collection of configurations as bit vectors, where true values indicate that specific orbitals are occupied.

Example

julia> bcs = BitConfigurations([[:a,:b,:c], [:x,:b,:c], [:a,:y,:c], [:a,:b,:z]])
6-orbital 4-configuration BitConfigurations

1: a b c
2: a -> x
3: b -> y
4: c -> z

julia> h = FieldFreeOneBodyHamiltonian()
ĥ₀

julia> Matrix(bcs, h)
4×4 SparseMatrixCSC{NBodyMatrixElement, Int64} with 10 stored entries:
(a|a) + (b|b) + (c|c)  (a|x)                  - (b|y)                (c|z)
(x|a)                  (b|b) + (c|c) + (x|x)  ⋅                      ⋅
- (y|b)                ⋅                      (a|a) + (c|c) + (y|y)  ⋅
(z|c)                  ⋅                      ⋅                      (a|a) + (b|b) + (z|z)
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EnergyExpressions.OrbitalsType
Orbitals(orbitals, overlaps, has_overlap, non_orthogonalities)

Structure storing a common set of orbitals, along with possible overlaps between them, in case of non-orthogonalities. has_overlap is a boolean matrix indicates if a pair of orbitals have overlap, either due to non-orthogonality or if they are the same orbital. non_orthogonalities is a boolean vector that indicates if a specific orbital is non-orthogonal to any other orbital in the set of orbitals. This structure is used internally by BitConfigurations.

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EnergyExpressions.non_zero_cofactorsFunction
non_zero_cofactors(sd, N, i, j)

Find all non-zero cofactors of the orbital overlap matrix between the Slater determinants i & j of sd, generated when striking out N rows & columns. This routine is tailored towards the case when few non-orthogonalities are present, e.g. approximately proportional to the number of orbitals.

Non-orthogonality between spin-orbitals is handled by dividing the them into two subspaces:

1. The orthogonal spin-orbitals that are common to both Slater determinants (core orbitals),

2. All non-orthogonal orbitals, and the orbitals which differ between the Slater determinants (i.e. holes of i and particles of j).

The relative phase between the Slater determinants is determined by group 2 alone, by permuting the particles to the positions of the holes, we find this phase. We can then formally permute them together to a diagonal block at lower-right corner of the orbital overlap matrix without incurring a phase change, since we need to permute the same number of rows and columns. We thus get this structure:

                 ╷       ╷
│ 1 │   │
det(Sᵢⱼ) = (-)ᵏ │───┼───│
│   │ 2 │
╵       ╵

where k is decided by the permutation necessary to put the particles in the positions of the holes.

Obviously, the determinant of the orbital matrix is now given by det(Sᵢⱼ) = (-)ᵏ*det(2), since we trivially have det(1)==1.

Depending on the rank of 2 (determined by the number of hole–particle pairs and which spin-orbitals are non-orthogonal), we need to strike out at least size(2,1)-rank(2) rows/columns from 2, and at most min(N,size(2,1)), i.e. for each value of n ∈ size(2,1)-rank(2):min(N,size(2,1)), we need to additionally strike out m = N - n rows from 1, but since the determinant of subspace 1 is unity, regardless of how many rows/columns we've stricken out, this is a trivial excercise. Of course, we also require that m ≤ size(1,1).

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