Electronic Integrals

The constructor functions listed here generally take the coefficients of electronic structure Hamiltonians as an input. Different flavors exist:

tril: these functions consume 1-dimensional arrays of flattened (generalized) triangular indices
full: these functions consume high-dimensional arrays
spin: these functions take separate arrays for the different spin species
sym: these functions take a single array for one spin species and infer the other spin species

qf_ferm_op_from_1body_tril_spin_sym

QfFermionOperator *qf_ferm_op_from_1body_tril_spin_sym(double *one_body_a, uint32_t norb)

Constructs an operator from spin-symmetric triangular 1-body integrals.

The resulting operator is defined by

\sum_i c^\alpha_{ii} (a^\dagger_i a_i + a^\dagger_{i+n} a_{i+n}) + \sum_{i \lt j} c^\alpha_{ij} (a^\dagger_i a_j + a^\dagger_j a_i + a^\dagger_{i+n} a_{j+n} + a^\dagger_{j+n} a_{i+n})

where $c^\alpha$ are the integral coefficients stored in one_body_a, $i$ and $j$ are the indices expanded from the triangular index $ij$ which indexes the array, and $n$ is the number of orbitals, norb.

1int norb = 2;
2double one_body_a[3] = {1.0, 2.0, 3.0};
3QfFermionOperator *op = qf_ferm_op_from_1body_tril_spin_sym(one_body_a, norb);

Parameters

one_body_a – a 1-dimensional array of length norb * (norb + 1) / 2 storing the 1-body electronic integral coefficients of the alpha-spin species, as a flattened triangular matrix.
norb – the number of orbitals.

Returns

The 1-body component of the electronic structure Hamiltonian as defined above.

qf_ferm_op_from_1body_tril_spin

QfFermionOperator *qf_ferm_op_from_1body_tril_spin(double *one_body_a, double *one_body_b, uint32_t norb)

Constructs an operator from separate spin-species triangular 1-body integrals.

The resulting operator is defined by

\sum_i c^\alpha_{ii} a^\dagger_i a_i + c^\beta_{ii} a^\dagger_{i+n} a_{i+n} + \sum_{i \lt j} c^\alpha_{ij} (a^\dagger_i a_j + a^\dagger_j a_i) + c^\beta_{ij} (a^\dagger_{i+n} a_{j+n} + a^\dagger_{j+n} a_{i+n})

where $c^\alpha$ ( $c^\beta$ ) are the integral coefficients stored in one_body_a (one_body_b, resp.), $i$ and $j$ are the indices expanded from the triangular index $ij$ which indexes the arrays, and $n$ is the number of orbitals, norb.

1int norb = 2;
2double one_body_a[3] = {1.0, 2.0, 3.0};
3double one_body_b[3] = {-1.0, -2.0, -3.0};
4QfFermionOperator *op = qf_ferm_op_from_1body_tril_spin(one_body_a, one_body_b, norb);

Parameters

one_body_a – a 1-dimensional array of length norb * (norb + 1) / 2 storing the 1-body electronic integral coefficients of the alpha-spin species, as a flattened triangular matrix.
one_body_b – a 1-dimensional array of length norb * (norb + 1) / 2 storing the 1-body electronic integral coefficients of the beta-spin species, as a flattened triangular matrix.
norb – the number of orbitals.

Returns

The 1-body component of the electronic structure Hamiltonian as defined above.

qf_ferm_op_from_2body_tril_spin_sym

QfFermionOperator *qf_ferm_op_from_2body_tril_spin_sym(double *two_body_aa, uint32_t norb)

Constructs an operator from spin-symmetric triangular 2-body integrals.

The resulting operator is defined by

\sum_{ijkl} \frac{1}{2} c^{\alpha\alpha}_{ijkl} \sum_{(i,j,k,l) \in \mathcal{P}(ijkl)} (a^\dagger_i a^\dagger_k a_l a_j + a^\dagger_{i+n} a^\dagger_k a_l a_{j+n} + a^\dagger_i a^\dagger_{k+n} a_{l+n} a_j + a^\dagger_{i+n} a^\dagger_{k+n} a_{l+n} a_{j+n})

where $c^{\alpha\alpha}$ are the integral coefficients stored in two_body_aa, $ijkl$ is the running index of the array, $\mathcal{P}$ generates the unique permutations of the 4-index $(i,j,k,l)$ (see below), and $n$ is the number of orbitals, norb.

1int norb = 2;
2double two_body_aa[6] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0};
3QfFermionOperator *op = qf_ferm_op_from_2body_tril_spin_sym(two_body_aa, norb);

Note

two_body_aa is an S8-fold symmetric array. That means, it is the flattened lower-triangular data of a matrix of shape (npair, npair), where npair = (norb * (norb + 1) // 2. This in turn is the lower-triangular data of the 4-dimensional array of shape (norb, norb, norb, norb). Therefore, $\mathcal{P}$ above expands the flattened index $ijkl$ into all index permutations $(i,j,k,l)$ that index this 4-dimensional array.

Parameters

two_body_aa – a 1-dimensional array of the S8-fold symmetric 2-body electronic integral coefficients of the alpha/alpha-spin species, as a flattened array.
norb – the number of orbitals.

Returns

The 2-body component of the electronic structure Hamiltonian as defined above.

qf_ferm_op_from_2body_tril_spin

QfFermionOperator *qf_ferm_op_from_2body_tril_spin(double *two_body_aa, double *two_body_ab, double *two_body_bb, uint32_t norb)

Constructs an operator from separate spin-species triangular 2-body integrals.

The resulting operator is defined by

\sum_{ijkl} \frac{1}{2} \sum_{(i,j,k,l) \in \mathcal{P}(ijkl)} c^{\alpha\alpha}_{ijkl} a^\dagger_i a^\dagger_k a_l a_j + c^{\beta\beta}_{ijkl} a^\dagger_{i+n} a^\dagger_{k+n} a_{l+n} a_{j+n} + \sum_{ijkl} \frac{1}{2} \sum_{(i,j,k,l) \in \mathcal{P'}(ijkl)} c^{\alpha\beta}_{ijkl} a^\dagger_{i+n} a^\dagger_k a_l a_{j+n} + c^{\alpha\beta}_{ijkl} a^\dagger_i a^\dagger_{k+n} a_{l+n} a_j +

where $c^{\alpha\alpha}$ ( $c^{\alpha\beta}$ , $c^{\beta\beta}$ ) are the integral coefficients stored in two_body_aa (two_body_ab, two_body_bb, resp.), $ijkl$ is the running index of the array, $\mathcal{P}$ ( $\mathcal{P'}$ ) generates the unique permutations of the 4-index $(i,j,k,l)$ (see below), and $n$ is the number of orbitals, norb.

1int norb = 2;
2double two_body_aa[6] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0};
3double two_body_ab[9] = {11.0, 12.0, 13.0, 14.0, 15.0,
4                         16.0, 17.0, 18.0, 19.0};
5double two_body_bb[6] = {-1.0, -2.0, -3.0, -4.0, -5.0, -6.0};
6QfFermionOperator *op = qf_ferm_op_from_2body_tril_spin(
7    two_body_aa, two_body_ab, two_body_bb, norb);

Note

two_body_aa and two_body_bb are a S8-fold symmetric arrays. That means, they are the flattened lower-triangular data of matrices of shape (npair, npair), where npair = (norb * (norb + 1) // 2. These in turn are the lower-triangular data of the 4-dimensional arrays of shape (norb, norb, norb, norb). Therefore, $\mathcal{P}$ above expands the flattened index $ijkl$ into all index permutations $(i,j,k,l)$ that index these 4-dimensional arrays.

However, two_body_ab is only S4-fold symmetric. Thus, it contains the full data of the (npair, npair) matrix (but still in flattened form). $\mathcal{P'}$ performs the corresponding index expansion. (In the definition above, we reused the index $ijkl$ as an abuse of notation.)

Parameters

two_body_aa – a 1-dimensional array of the S8-fold symmetric 2-body electronic integral coefficients of the alpha/alpha-spin species, as a flattened array.
two_body_ab – a 1-dimensional array of the S4-fold symmetric 2-body electronic integral coefficients of the alpha/beta-spin species, as a flattened array.
two_body_bb – a 1-dimensional array of the S8-fold symmetric 2-body electronic integral coefficients of the beta/beta-spin species, as a flattened array.
norb – the number of orbitals.

Returns

The 2-body component of the electronic structure Hamiltonian as defined above.