Wavefunction File formats#

The trial wavefunction in AFQMC is typically a linear combination of Slater determinants,

\[|\Psi_\mathrm{T} \rangle = \sum^{N_\mathrm{det}}_n C_n | \Phi_n \rangle\]

where \(C_n\) is a complex-valued coefficient, and \(|\Phi_n\rangle\) are Slater determinants which are not necessarily orthogonal to each other. Of course, each Slater determinant consists of some set of single-particle orbitals, \(\{ \psi_p \}\), such that,

\[\psi_{p} = \sum_i \bar{C}_{ip} \phi_i\]

where \(\{\phi_i\}\) are the chosen orthonormal basis set orbitals. Slater determinants can either be represented explicitly as a Slater matrix,

\[\begin{split}\Phi = \begin{bmatrix} \bar{C}_{00} &\bar{C}_{01} & \bar{C}_{02} & \dots & \bar{C}_{0N} \\ \bar{C}_{10} & \bar{C}_{11} & \bar{C}_{12} & \dots & \bar{C}_{1N} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \bar{C}_{M0} & \bar{C}_{M1} & \bar{C}_{M2} & \dots & \bar{C}_{MN} \end{bmatrix}\end{split}\]

where \(M,N\) are the number of basis functions and electrons, respectively, or in terms of an “occupation vector”, which is simply a list of orbital indices which should be occupied. See the SAFIRE tutorials for tutorials and examples on generating and writing trial wavefunctions.

SAFIRE implements two basic types of trial wavefunction:

  1. “particle-hole” multi-Slater determinant (ph-mSD) trial wavefunctions which is a configuration interaction-like wavefunction where \(\langle \Phi_n |\Phi_m\rangle = \delta_{nm} \forall n,m\). In this case, the Slater determinants are specified in terms of a set of orbitals, and a list of \(C_n\) and corresponding occupancy vectors. See the HDF5 file format description in PHMSD for more.

  2. non-orthogonal multi-slater Determinant (NOMSD) trial wavefunction where strictly \(\langle \Phi_n |\Phi_m\rangle \neq 0 \forall n,m\). In this case, the Slater determinants are specified as a list of \(C_n\) and corresponding Slater matrices. See the HDF5 file format description in NOMSD for more.

We’ll explore each of these wavefunction types in more detail below.

Note

Using the ph-mSD trial wavefunction is recommended when using a large number of determinants since it is significantly more memory efficient than the NOMSD trial wavefunction, and since, in some cases, fast Woodbury updates can be applied.

Warning

When using NOMSD trial wavefunctions, it is important that each Slater determinant has a non-zero overlap with all other Slater determinants to avoid singular values. Use ph-mSD for orthogonal expansions of Slater determinants.

AFQMC allows for two types of multi-determinant trial wavefunctions: non-orthogonal multi Slater determinants (NOMSD) or SHCI/CASSCF style particle-hole multi Slater determinants (PHMSD).

NOMSD#

h5dump -n wfn.h5

HDF5 "wfn.h5" {
    FILE_CONTENTS {
        group      /
        group      /Wavefunction
        group      /Wavefunction/NOMSD
        dataset    /Wavefunction/NOMSD/Psi0_alpha
        dataset    /Wavefunction/NOMSD/Psi0_beta
        group      /Wavefunction/NOMSD/PsiT_0
        dataset    /Wavefunction/NOMSD/PsiT_0/data_
        dataset    /Wavefunction/NOMSD/PsiT_0/dims
        dataset    /Wavefunction/NOMSD/PsiT_0/jdata_
        dataset    /Wavefunction/NOMSD/PsiT_0/pointers_begin_
        dataset    /Wavefunction/NOMSD/PsiT_0/pointers_end_
        group      /Wavefunction/NOMSD/PsiT_1
        dataset    /Wavefunction/NOMSD/PsiT_1/data_
        dataset    /Wavefunction/NOMSD/PsiT_1/dims
        dataset    /Wavefunction/NOMSD/PsiT_1/jdata_
        dataset    /Wavefunction/NOMSD/PsiT_1/pointers_begin_
        dataset    /Wavefunction/NOMSD/PsiT_1/pointers_end_
        dataset    /Wavefunction/NOMSD/ci_coeffs
        dataset    /Wavefunction/NOMSD/dims
    }
}

Note that the \(\alpha\) components of the trial wavefunction are stored under PsiT_{2n} and the \(\beta\) components are stored under PsiT_{2n+1}.

  • /Wavefunction/NOMSD/Psi0_alpha \([M,N_\alpha]\) dimensional array \(\alpha\) component of initial walker wavefunction.

  • /Wavefunction/NOMSD/Psi0_beta \([M,N_\beta]\) dimensional array for \(\beta\) initial walker wavefunction.

  • /Wavefunction/NOMSD/PsiT_{2n}/data_ Array of length \(nnz\) containing non-zero elements of \(n\)-th \(\alpha\) component of trial wavefunction walker wavefunction. Note the conjugate transpose of the Slater matrix is stored.

  • /Wavefunction/NOMSD/PsiT_{2n}/dims Array of length 3 containing \([M,N_{\alpha},nnz]\) where \(nnz\) is the number of non-zero elements of this Slater matrix

  • /Wavefunction/NOMSD/PsiT_{2n}/jdata_ CSR indices array.

  • /Wavefunction/NOMSD/PsiT_{2n}/pointers_begin_ CSR format begin index pointer array.

  • /Wavefunction/NOMSD/PsiT_{2n}/pointers_end_ CSR format end index pointer array.

  • /Wavefunction/NOMSD/ci_coeffs \(N_D\) length array of ci coefficients. Stored as complex numbers.

  • /Wavefunction/NOMSD/dims Integer array of length 5 containing \([M,N_\alpha,N_\beta,\) walker_type \(,N_D]\)

PHMSD#

h5dump -n wfn.h5

HDF5 "wfn.h5" {
    FILE_CONTENTS {
        group      /
        group      /Wavefunction
        group      /Wavefunction/PHMSD
        dataset    /Wavefunction/PHMSD/Psi0_alpha
        dataset    /Wavefunction/PHMSD/Psi0_beta
        dataset    /Wavefunction/PHMSD/ci_coeffs
        dataset    /Wavefunction/PHMSD/dims
        dataset    /Wavefunction/PHMSD/occs
        dataset    /Wavefunction/PHMSD/type
    }
}

  • /Wavefunction/NOMSD/Psi0_alpha \([M,N_\alpha]\) dimensional array \(\alpha\) component of initial walker wavefunction.

  • /Wavefunction/NOMSD/Psi0_beta \([M,N_\beta]\) dimensional array for \(\beta\) initial walker wavefunction.

  • /Wavefunction/PHMSD/ci_coeffs \(N_D\) length array of ci coefficients. Stored as complex numbers.

  • /Wavefunction/PHMSD/dims Integer array of length 5 containing \([M,N_\alpha,N_\beta,\) walker_type \(,N_D]\)

  • /Wavefunction/PHMSD/occs Integer array of length \((N_\alpha+N_\beta)*N_D\) describing the determinant occupancies. For example if \((N_\alpha=N_\beta=2)\) and \(N_D=2\), \(M=4\), and if \(|\Psi_\mathrm{T}\rangle = |0,1\rangle|0,1\rangle + |0,1\rangle|0,2\rangle>\) then occs = \([0, 1, 4, 5, 0, 1, 4, 6]\). Note that \(\beta\) occupancies are displaced by \(M\).

  • /Wavefunction/PHMSD/type integer 0/1. 1 implies trial wavefunction is written in different basis than the underlying basis used for the integrals. If so a matrix of orbital coefficients is required to be written in the NOMSD format. If 0 then assume wavefunction is in same basis as integrals.

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