Hamiltonian File formats

The Hamiltonian is, of course, one of the key inputs to SAFIRE. SAFIRE is aware only of the generic class of Hamiltonian, so that it can take advantage of class-specific algorithms while remaining general. Python-based tools to generate Hamiltonians, among other inputs, are provided in the afqmctools module. See the afqmctools documentation afqmctools and the tutorials Tutorials for more details.

Several generic classes of Hamiltonians are implemented and described in detail below.

1. Dense Cholesky

2. K-Point Factorized (KP)

3. Lattice Model Hamiltonian

4. THC Hamiltonian

Dense Cholesky

A common Hamiltonian type that is implemented in SAFIRE is based on the modified-Cholesky factorization [ADVFerre+09, BL77, KdMerasP03, PKVZ11, PZK13] of the ERI tensor:

(2)\[v_{pqrs} = V_{(pr),(sq)} \approx \sum_n^{N_{\mathrm{chol}}} L_{pr,n} L^{*}_{sq,n},\]

where the sum is truncated at \(N_{\mathrm{chol}} = x_c M\), \(x_c\) is typically between \(5\) and \(10\), \(M\) is the number of basis functions and we have assumed that the single-particle orbitals are in general complex. The storage requirement is thus naively \(\mathcal{O}(M^3)\). Note we follow the usual definition of \(v_{pqrs} = \langle pq | rs \rangle = (pr|qs)\). With this form of factorization, SAFIRE allows for the integrals to be stored in either dense or sparse format.

The dense case is the simplest and is only implemented for Hamiltonians with real integrals (and basis functions, i.e. not the homogeneous electron gas which has complex orbitals but real integrals). The file format is given as follows:

Listing 8 Sample Dense Cholesky AFQMC Hamiltonian.

$ h5dump -n afqmc.h5
HDF5 "afqmc.h5" {
    FILE_CONTENTS {
        group      /
        group      /Hamiltonian
        dataset    /Hamiltonian/dims
        dataset    /Hamiltonian/Energies
        dataset    /Hamiltonian/hcore
        group      /Hamiltonian/DenseFactorized
        dataset    /Hamiltonian/DenseFactorized/L
    }
}

Required datasets:

• /Hamiltonian/dims Descriptor array of length 8 containing \([0,0,0,M,N_\alpha,N_\beta,0,N_\mathrm{chol}]\). \(M\) is the number of basis functions, \(N_\alpha\) and \(N_\beta\) are the numbers of spin-up and spin-down electrons, and \(N_\mathrm{chol}\) is the number of Cholesky vectors.

• /Hamiltonian/Energies Array containing \([E_{II}, E_{\mathrm{core}}]\). \(E_{II}\) should contain ion-ion repulsion energy and any additional constant terms which have to be added to the total energy. \(E_{\mathrm{core}}\) is deprecated and not used.

• /Hamiltonian/hcore One-body Hamiltonian matrix elements. For real integrals, dimensions are \([M_{spin}, M]\) where \(M_{spin}\) depends on the spin symmetry: \(M\) for closed-shell, \(2M\) for collinear, or \(2M\) for noncollinear systems. For complex integrals, an additional dimension of size 2 is added to store real and imaginary parts separately.

• /Hamiltonian/DenseFactorized/L The Cholesky decomposition tensor \(L_{pr,n}\). Dimensions are \([M \times M, N_\mathrm{chol}]\) where the first dimension represents the flattened orbital pair indices \((p,r)\) and the second dimension represents the Cholesky vector index \(n\). Only real-valued Cholesky vectors are supported for the Dense Cholesky format.

Storage conventions:

• The Dense Cholesky format only supports real-valued Cholesky vectors.

• Complex one-body Hamiltonians should be stored with an additional dimension of size 2, where array[..., 0] contains real parts and array[..., 1] contains imaginary parts.

• The Cholesky tensor \(L\) is stored in column-major order with orbital pairs \((p,r)\) flattened into a single index.

• For collinear systems, the one-body Hamiltonian should have the spin-up block in the upper half and spin-down block in the lower half.

See also

See the Writing a Hamiltonian tutorial from the SAFIRE for Quantum Chemistry tutorials.

K-Point Factorized (KP)

We have implemented an explicitly \(k\)-point dependent factorization for periodic systems [MZM20, MZC19]

(3)\[(\textbf{k}_p p \textbf{k}_r r| \textbf{k}_q q \textbf{k}_s s) = \sum_n L^{\textbf{Q},\textbf{k}}_{pr,n} {L^{\textbf{Q},\textbf{k}'}_{sq,n}}^{*}\]

where \(\textbf{k}\), \(\textbf{k}'\) and \(\textbf{Q}\) are vectors in the first Brillouin zone. The one-body Hamiltonian is block diagonal in \(\textbf{k}\) and in (3) we have used momentum conservation \((\textbf{k}_p - \textbf{k}_r + \textbf{k}_q - \textbf{k}_s) = \textbf{G}\) with \(\textbf{G}\) being some vector in the reciprocal lattice of the simulation cell. The convention for the Cholesky matrix \(L^{\textbf{Q},\textbf{k}}_{pr,\gamma}\) is as follows: \(\textbf{k}_r = \textbf{k}_p - \textbf{Q}\), so the vector \(\textbf{k}\) labels the k-point of the first band index, \(\textit{p}\), while the k-point vector of the second band index, \(\textit{r}\), is given by \(\textbf{k} - \textbf{Q}\). Electron repulsion integrals at different \(\textbf{Q}\) vectors are zero by symmetry, resulting in a reduction in the number of required \(\mathbf{Q}\) vectors. For certain \(\textbf{Q}\) vectors that satisfy \(\textbf{Q} \ne -\textbf{Q}\) (this is not satisfied at the origin and at high symmetry points on the edge of the 1BZ), we have \({L^{\textbf{Q},\textbf{k}}_{sq,\gamma}}^{*} = {L^{-\textbf{Q},\textbf{k}-\textbf{Q}}_{qs,\gamma}}\), which requires us to store Cholesky vectors for either one of the \((\textbf{Q},-\textbf{Q})\) pair, but not both.

In what follows let \(m_{\mathbf{k}}\) denote the number of basis functions for basis functions of a given \(k\)-point (these can in principle differ for different \(k\)-points due to linear dependencies), \(n^{\alpha}_{\mathbf{k}}\) the number of \(\alpha\) electrons in a given \(k\)-point and \(n_{\mathrm{chol}}^{\mathbf{Q}_n}\) the number of Cholesky vectors for momentum transfer \(\mathbf{Q}_n\). The file format for this factorization is as follows (for a \(2\times2\times2\) \(k\)-point mesh, for denser meshes generally there will be far fewer symmetry inequivalent momentum transfer vectors than there are \(k\)-points):

Listing 9 Sample Dense \(k\)-point dependent Cholesky SAFIRE Hamiltonian.

$ h5dump -n afqmc.h5
HDF5 "afqmc.h5" {
    FILE_CONTENTS {
        group      /
        group      /Hamiltonian
        group      /Hamiltonian/KPFactorized
        dataset    /Hamiltonian/KPFactorized/L0
        dataset    /Hamiltonian/KPFactorized/L1
        dataset    /Hamiltonian/KPFactorized/L2
        dataset    /Hamiltonian/KPFactorized/L3
        dataset    /Hamiltonian/KPFactorized/L4
        dataset    /Hamiltonian/KPFactorized/L5
        dataset    /Hamiltonian/KPFactorized/L6
        dataset    /Hamiltonian/KPFactorized/L7
        dataset    /Hamiltonian/NCholPerKP
        dataset    /Hamiltonian/MinusK
        dataset    /Hamiltonian/NMOPerKP
        dataset    /Hamiltonian/QKTok2
        dataset    /Hamiltonian/H1_kp0
        dataset    /Hamiltonian/H1_kp1
        dataset    /Hamiltonian/H1_kp2
        dataset    /Hamiltonian/H1_kp3
        dataset    /Hamiltonian/H1_kp4
        dataset    /Hamiltonian/H1_kp5
        dataset    /Hamiltonian/H1_kp6
        dataset    /Hamiltonian/H1_kp7
        dataset    /Hamiltonian/ComplexIntegrals
        dataset    /Hamiltonian/KPoints
        dataset    /Hamiltonian/dims
        dataset    /Hamiltonian/Energies
    }
}

• /Hamiltonian/KPFactorized/L[n] This series of datasets store elements of the Cholesky tensors \(L[\mathbf{Q}_n,\mathbf{k},pr,n]\). Each data set is of dimension \([N_k,m_{\mathbf{k}}\times m_{\mathbf{k}'},n^{\mathbf{Q}_n}_\mathrm{chol}]\), where, again, \(k\) is the \(k\)-point associated with basis function \(p\), the \(k\)-point of basis function \(r\) is defined via the mapping QKtok2.

• /Hamiltonian/NCholPerKP \(N_k\) length array giving number of Cholesky vectors per \(k\)-point.

• /Hamiltonian/MinusK: \(N_k\) length array mapping a \(k\)-point to its inverse: \(\mathbf{k}_i+\)MinusK[i] \(= \mathbf{0} \mod \mathbf{G}\).

• /Hamiltonian/NMOPerKP: \(N_k\) length array listing number of basis functions per \(k\)-point.

• /Hamiltonian/QKTok2: \([N_k,N_k]\) dimensional array. QKtok2[i,j] yields the \(k\) point index satisfying \(\mathbf{k}=\mathbf{Q}_i-\mathbf{k}_j+\mathbf{G}\).

• /Hamiltonian/dims: Descriptor array of length 8 containing \([0,0,0,M,N_\alpha,N_\beta,0,0]\).

• /Hamiltonian/H1_kp[n] Contains the \([m_{\mathbf{k}_n},m_{\mathbf{k}_n}]\) dimensional one-body Hamiltonian matrix elements \(h_{(\mathbf{k}_{n}p)(\mathbf{k}_{n}q)}\).

• /Hamiltonian/ComplexIntegrals Length 1 array that specifies if integrals are complex valued. 1 for complex integrals, 0 for real integrals.

• /Hamiltonian/KPoints \([N_k,3]\) Dimensional array containing \(k\)-points used to sample Brillouin zone.

• /Hamiltonian/dims Descriptor array of length 8 containing \([0,0,N_k,M,N_\alpha,N_\beta,0,N_\mathrm{nchol}]\). Note that \(M\) is the total number of basis functions, i.e. \(M=\sum_\mathbf{k} m_\mathbf{k}\), and likewise for the number of electrons.

• /Hamiltonian/Energies Array containing \([E_{II}, E_{\mathrm{core}}]\). \(E_{II}\) should contain ion-ion repulsion energy and any additional constant terms which have to be added to the total energy (such as the electron-electron interaction Madelung contribution of \(\frac{1}{2} N \xi )\). \(E_{\mathrm{core}}\) is deprecated and not used.

Complex integrals should be written as an array with an additional dimension, e.g., a 1D array should be written as a 2D array with array_hdf5[:,0]=real(1d_array) and array_hdf5[:,1]=imag(1d_array). The functions afqmctools.utils.misc.from_complex and afqmctools.utils.misc.to_complex can be used to transform from the internal complex format to complex valued numpy arrays of the appropriate shape and vice versa.

See also

See the following from the SAFIRE for ab initio solids tutorials.

• (under construction) Writing a Hamiltonian with CoQuí

Lattice Model Hamiltonian

General multi-band, Hubbard-Kanamori (HK) lattice model Hamiltonians are implemented. The HK Hamiltonian has the general form,

\[\begin{split}\hat{H} = \sum_{ij,\sigma \sigma'}t^{\sigma \sigma'}_{ij}\hat{c}^\dagger_{i\sigma}\hat{c}_{j\sigma'} + \sum_{i} U_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} \\ + \sum_{i<j} U_{ij}^1 (\hat{n}_{i\uparrow} \hat{n}_{j\downarrow} + \hat{n}_{i\downarrow} \hat{n}_{j\uparrow} ) + \sum_{i<j} U_{ij}^2 (\hat{n}_{i\uparrow} \hat{n}_{j\uparrow} + \hat{n}_{i\downarrow} \hat{n}_{j\downarrow} ) \\ + \sum_{i<j} J_{ij} ( \hat{c}^\dagger_{i\uparrow}\hat{c}^\dagger_{j\downarrow}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow} +\hat{c}^\dagger_{i\uparrow}\hat{c}^\dagger_{i\downarrow}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow} +\hat{c}^\dagger_{j\uparrow}\hat{c}^\dagger_{i\downarrow}\hat{c}_{j\downarrow}\hat{c}_{i\uparrow} +\hat{c}^\dagger_{j\uparrow}\hat{c}^\dagger_{j\downarrow}\hat{c}_{i\downarrow}\hat{c}_{i\uparrow} ),\end{split}\]

where \(i\),\(j\) are combined lattice and band indices, \(\hat{c}^\dagger_{i\sigma}\), \(\hat{c}_{j\sigma'}\) create/annihilate and electron on the site (and band) corresponding to \(i\)/\(j\) with spin \(\sigma\)/\(\sigma'\), \(\hat{n}_{i\sigma}\) is the number operator, \(t^{\sigma \sigma'}_{ij}\) includes all one-body terms (\(n^{th}\)-order neighbor hopping, spin-orbit coupling, etc.), \(U\) is the traditional on-site hubbard interaction, \(U^1\) is a density-density interaction, \(U^2\) is a spin-spin interaction, and \(J\) is a Hund’s coupling term. \(U^1\), \(U^2\), and \(J\) are typically only non-zero between bands on the same lattice site, but are by no means limited in this way.

Each individual term is optional (including the hopping term); however, including at least one interaction term is required. Model Hamiltonians are saved in HDF5 using the following format.

Listing 10 Sample Hubbard-Kanamori SAFIRE Hamiltonian.

$ h5dump -n afqmc.h5
  HDF5 "afqmc.h5" {
    FILE_CONTENTS {
        group      /
        group      /Hamiltonian
        dataset    /Hamiltonian/Energies
        group      /Hamiltonian/ModelHamiltonian
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_0
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/model_type
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/spin_type
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij/data_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij/dims
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij/jdata_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij/pointers_begin_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_0/tij/pointers_end_
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_1
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij/data_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij/dims
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij/jdata_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij/pointers_begin_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/Jij/pointers_end_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/hst_type
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/model_type
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_1/spin_type
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_2
        group      /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij/data_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij/dims
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij/jdata_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij/pointers_begin_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/Uij/pointers_end_
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/hst_type
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/model_type
        dataset    /Hamiltonian/ModelHamiltonian/ModelComponent_2/spin_type
        dataset    /Hamiltonian/ModelHamiltonian/maximum_connectivity
        dataset    /Hamiltonian/ModelHamiltonian/number_of_components
        dataset    /Hamiltonian/dims
        dataset    /Hamiltonian/spin_type
  }
}

• /Hamiltonian/Energies: Array containing \([E_{II}, E_{\mathrm{core}}]\). \(E_{II}\) should contain ion-ion repulsion energy and any additional constant terms which have to be added to the total energy (such as the electron-electron interaction Madelung contribution of \(\frac{1}{2} N \xi )\). \(E_{\mathrm{core}}\) is deprecated and not used. In the context of lattice models, both \(E_{II}\) and \(E_{\mathrm{core}}\) while generally be set to 0.0.

• /Hamiltonian/dims: Descriptor array of length 8 containing \([0,0,0,M,M,0,0,0]\). Note that \(M\) is the total number of basis functions, i.e. number of sites * number of bands

• /Hamiltonian/spin_type: Is a string containing the spin symmetry type of the Hamiltonian. possible values are “collinear”, or “noncollinear”.

• /Hamiltonian/ModelHamiltonian/maximum_connectivity: A scalar integer containing the maximum number of possible connections in the Hamiltonian. If in doubt, the number of non-zero elements can be safely used here.

• /Hamiltonian/ModelHamiltonian/number_of_components: A scalar integer containing the number of “components” (i.e. terms) in the Hamiltonian. This should be consistent with the number of :math:` /Hamiltonian/ModelHamiltonian/ModelComponent_n/` groups, where \(n\) is replaced with an integer.

• /Hamiltonian/ModelHamiltonian/ModelComponent_n: (where \(n\) are replaced by a specific integer index) are HDF5 groups which contain a Hamiltonian component (i.e. a term). Each component consists of some metadata and a compressed sparse row (CSR) matrix representation of that Hamiltonian component. Each group has the following datasets and groups.

  • /spin_type: a string containing the spin symmetry type of the component. Possible values are “collinear”, or “noncollinear”.

  • /model_type: a string describing the type of model term. Possible values are: “one_body”, “hubbard_u”, and “hubbard_j”.

  • a matrix group. The name of the group depends on the /model_type. Possible values are: /tij/ ("one_body"), /Uij/ ("hubbard_u"), and /Jij/ ("hubbard_j"). Conventions for the shape of each matrix are described below. The matrix group will always have the following datasets:

  • dims: Descriptor array of length 3 containing \([M_1,M_2,N_{nz}]\). where \(M_1\) and \(M_2\) are the dimensions of the CSR matrix, and \(N_{nz}\) is the number of non-zero entries.

  • data_: an array containing the non-zero matrix elements.

  • jdata_: an array containing the column indices of each non-zero matrix element.

  • pointers_begin_: an array containing “pointers” to the beginning of each row within data_ / jdata_.

  • pointers_end_: an array containing “pointers” to the end of each row within data_ / jdata_.

Model Component Conventions

The following Conventions apply to the CSR matrix representations of each model component.

• both real- and complex-valued matrices are supported; however, all components must have the same type. i.e. if the hopping matrix is complex-valued, due to a twist, all interaction terms must also be complex-valued even of the imaginary part is identically zero.

• for all interaction terms except for on-site, on-band Hubbard U, we take the convention that \(i < j\) as shown in the general Hubbard-Kanamori Hamiltonian above. If entries with \(i >= j\) are supplied, they will be ignored with a warning.

• For /tij/, if \(M\) is the total basis set size, including bands and sites, the CSR matrix may have dimension \(M x M\), \(2M x M\), or \(2M x 2M\) corresponding to "closed", "collinear", and "noncollinear" spin symmetry, respectively. For a "closed" hopping matrix, the same hopping will be applied to both spin channels. For a "collinear" hopping matrix, the upper half of /tij/ will applied to the spin up channel, and the lower half to the spin-down channel. For a "noncollinear" hopping matrix, /tij/ is explicitly represented in a spin-orbital basis and is applied as is. We note that /tij/ may contain arbitrary one-body terms if desired; for example, a pinning field.

  the CSR matrix may have dimension \(M x M\), or \(2M x M\). If the dimension is \(2M x M\), then the upper half of the array is interpreted as \(U_{ii}+U1_{ij}\) with \(i,j\) running over all basis set functions, and the lower half is interpreted as \(U2_{ij}`with :math:`i,j\) running over all basis set functions. If \(U2_{ij}\) is desired without \(U_{ii}\) or \(U1_{ij}\), the matrix must still have dimension \(2M x M\), but the upper half should have no non-zero elements. If dimension is \(M x M\), then the CSR matrix is interpreted as \(U_{ii}+U1_{ij}\).

• for /Jij, the CSR matrix must have dimension \(M x M\) and is interpreted as the the \(J_{ij}\) from the Hubbard-Kanamori term.

The other groups and datasets represent the Hamiltonian “components” (i.e. the terms which are included in the specific Hamiltonian given to AFQMC).

See also

See the following from the SAFIRE for lattice models tutorials.

• Setting up a Lattice

• Building and writing a Hamiltonian

• Finer control of the Hamiltonian

Tensor Hyper Contraction (THC)

The Tensor Hyper-Contraction (THC) approach is an efficient method for representing two-electron integrals that exploits the low-rank structure of the electron repulsion integral (ERI) tensor. THC factorizes the four-center electron repulsion integrals into products involving lower-dimensional tensors, providing significant computational savings for AFQMC calculations.

THC represents the ERI tensor as:

(4)\[v_{pqrs} \approx \sum_{u}^{N_u} \sum_{v}^{N_v} X_{pu} L_{uv} X_{rv}^*\]

where \(X_{pu}\) are collocation matrices that map orbitals to interpolation points, \(L_{uv}\) is the factorized Coulomb matrix at interpolation points, \(N_u\) and \(N_v\) are the numbers of interpolation points, and the approximation quality depends on the choice of interpolation points and the rank of the factorization.

The THC format supports two main variants:

Option 1: Direct THC factorization

Uses the factorized Coulomb matrix \(L_{uv}\) directly and reuses the same collocation matrix for both electron coordinates.

Option 2: Half-rotated THC factorization

Uses a half-rotated Coulomb matrix and separate collocation matrices that have been pre-rotated to a specific orbital space.

THC Hamiltonians are stored in HDF5 using the following format:

Listing 11 Sample THC AFQMC Hamiltonian.

$ h5dump -n afqmc.h5
HDF5 "afqmc.h5" {
    FILE_CONTENTS {
        group      /
        group      /System
        dataset    /System/H0
        dataset    /System/atomic_positions
        dataset    /System/lattice_vectors
        [additional system datasets...]
        group      /Interaction
        dataset    /Interaction/factorized_coulomb_matrix
        dataset    /Interaction/collocation_matrix
        dataset    /Interaction/coulomb_matrix [Option 1 only]
        dataset    /Interaction/half_rotated_coulomb_matrix [Option 2 only]
        dataset    /Interaction/collocation_matrix_half_rotated [Option 2 only]
    }
}

Required datasets:

• /System/H0 One-body Hamiltonian matrix elements. Dimensions are [nspin, nkpts, norb, norb, complex_dim] where complex_dim=2 for complex integrals and is omitted for real integrals. For molecular calculations, nkpts=1.

• /Interaction/factorized_coulomb_matrix The factorized Coulomb tensor \(L_{uv}\). Dimensions are [nq, nu, nv, complex_dim] where nq should be 1 for current implementation. For complex integrals, the last dimension has size 2 storing real and imaginary parts.

• /Interaction/collocation_matrix The collocation matrix \(X_{pu}\) mapping orbitals to interpolation points. Dimensions are [nspin, nkpts, norb, nu, complex_dim].

Option 1 additional dataset:

• /Interaction/coulomb_matrix The full Coulomb matrix \(M_{uv} = L_{uv} \cdot L_{uv}^{\dagger}\). This allows reuse of the same collocation matrix. Dimensions are [nq, nu, nu, complex_dim].

Option 2 additional datasets:

• /Interaction/half_rotated_coulomb_matrix A half-rotated version of the Coulomb matrix for specialized THC factorizations. Dimensions are [nq, nu_rot, nv_rot, complex_dim].

• /Interaction/collocation_matrix_half_rotated The corresponding half-rotated collocation matrix. Dimensions match the half-rotated coulomb matrix structure.

Storage conventions:

• Complex integrals should be stored with an additional dimension of size 2, where array[..., 0] contains real parts and array[..., 1] contains imaginary parts.

• For spin-collinear calculations with nspin=2, if only one spin component is provided in the file (nspin_in_file=1), it will be automatically duplicated for the second spin channel.

• The parameter cutoff_cholesky can be specified to control the accuracy of Cholesky-related operations (default: 1e-6).

The THC representation provides substantial memory and computational savings compared to storing the full four-index ERI tensor, especially for large systems where the number of interpolation points \(N_u \ll M^2\) (where \(M\) is the number of orbitals).

See also

See the following from the SAFIRE for ab initio solids tutorials.

• (under construction) Writing a Hamiltonian with CoQuí

Final Notes on Hamiltonians

The code CoQuí is able to generate and output both the K-Point Factorized, and THC Hamiltonian format directly.

Lattice model Hamiltonians can be generated using the afqmctools Python package which is included with SAFIRE. See the lattice model tutorials for more information on how to generate these Hamiltonians.

Finally, if using external tools to generate this file format, we provide a sanity checker script in utils/bin/test_afqmc_input.py which will raise errors if the format does not conform to what is being used internally.