Ab-initio quantities

Quantities input to CC-aims

The Fock matrix

The Fock matrix \(\mathbf{F}\) denotes the matrix-representation of single particle Hartree-Fock operator

\[ f_{i}(\mathbf{r}) = -\frac{1}{2}\nabla_{i}^2 - \sum_{I}\frac{Z_{I}}{r_{i}-r_{I}} + \hat{J} - \hat{K} \]

(\(\hat{J}\) and \(\hat{K}\) being the Coulomb and exchange operator), in the space of converged molecular orbitals (MO) or - in the periodic case - Bloch states (BS). Hence, the Fock matrix elements are of the form

\[ (\mathbf{F})_{jk} = \langle \psi_{j}^{MO}(\mathbf{r_{i}})|f_{i}|\psi_{j}^{MO}(\mathbf{r_{i}})\rangle \]

\[ (\mathbf{F})_{jk} = \langle \psi_{j}^{BS}(\mathbf{r_{i}})|f_{i}|\psi_{j}^{BS}(\mathbf{r_{i}})\rangle. \]

If the self-consistent calculation performed is a HF calculation, the Fock matrix becomes diagonal, featuring the single-particle HF-energies of the diagonal. If, however, the underlying SCF method is KS-DFT is not diagonal anymore.

CC4S only requires the Fock matrix if a KS-DFT calculation has been performed, as otherwise the information in the Fock matrix is already given by the energy eigenvalues themselves.

HF/KS-DFT energy eigenvalues and eigenvectors

Similar to the Fock matrix, the energy eigenvalues and eigenvectors of the SCF-calculation are part of the regular output of any established quantum chemistry code. The eigenvalues are the solutions of the generalized eigenvalue problem, which needs to be solved in the HF or KS-DFT method:

\[ \mathbf{H}\mathbf{C} = \mathbf{S}\mathbf{C}\mathbf{\epsilon}, \]

where \(\mathbf{H}\) is the Fock matrix in the representation of atomic orbitals (AO), \(\mathbf{C}\) the eigenvector matrix, \(\mathbf{S}\) the overlap matrix of the AOs and \(\mathbf{\epsilon}\) the diagonal energy eigenvalue matrix.

Fourier-transformed RI-expansion coefficents

The representation of a product of Bloch basis functions as an expansion of RI-basis functions is given by:

\[ \phi_{i\mathbf{k}}^{*}(\mathbf{r})\phi_{k\mathbf{q}}(\mathbf{r}) = \sum_{\mu} C_{ik}^{\mu}(\mathbf{k}, \mathbf{q})P_{\mu}^{\mathbf{q}-\mathbf{k}} \]

If a calculation with a non-local RI-scheme is requested by CC-aims, the RI-expansion coefficients \(C_{ik}^{\mu}(\mathbf{k}, \mathbf{q})\) need to be submitted directly to the interface.

However, in the case of the local RI-LVL scheme, it has been shown by Levchenko et al. that \(C_{ik}^{\mu}(\mathbf{k}, \mathbf{q})\) become separable in reciprocal space:

\[ C_{ik}^{\mu}(\mathbf{k}, \mathbf{q}) = C_{ik}^{\mu}(-\mathbf{k}, \mathbf{R_{0}}) + C_{ki}^{\mu}(\mathbf{q}, \mathbf{R_{0}}), \]

where

\[ C_{ik}^{\mu}(\mathbf{k}) = C_{ik}^{\mu}(\mathbf{k}, \mathbf{R_{0}}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\mathbf{R}}C_{i(\mathbf{R}) k(\mathbf{R_{0}})}^{\mu(\mathbf{R_{0}})} \]

Here, \(C_{i(\mathbf{R})k(\mathbf{R_{0}})}^{\mu(\mathbf{R_{0}})}\) denotes the expansion coefficient of the RI-basis function \(P_{\mu}(\mathbf{r})\) of the central unit cell (signified by the real-space lattice vector \(\mathbf{R_{0}}\)) in the representation of the product of the \(i\)-th atomic basis function in unit cell \(\mathbf{R}\) and the \(k\)-th atomic basis function in the central unit cell \(\mathbf{R_{0}}\). Due to the translational symmetry of the general, periodic case, it sufficies to only look at basis function overlap between the central unit cell and adjacent unit cells.

Hence, if starting from a RI-LVL scheme, it suffices to provide CC-aims with \(C_{ik}^{\mu}(\mathbf{k})\), which can save significant amounts of memory.

The auxiliary Coulomb matrix

Finally, CC-aims requires information of the interaction/overlap between RI-basis functions, which is provided by the auxiliary Coulomb matrix

\[ V_{\mu\nu}^{\mathbf{q}-\mathbf{k}} = \int d\mathbf{r} d\mathbf{r'} \frac{P_{\mu}^{\mathbf{q}}(\mathbf{r}) P_{\nu}^{\mathbf{k}}(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}. \]

Quantities output by CC-aims

The purpose of CC-aims is to calculate the input quantities of CC4S using results obtained from a SCF calculation of an ab-initio quantum chemistry code, FHI-aims in particular. In addition to the Fock matrix and the SCF-eigenvalues, the quantities, calculated and/or output by CC-aims are:

Spin eigenvalues

If a spin-polarized SCF-calculation was performed, CC4S requires the \(S_z\)-eigenvalues as multiples of \(\hbar\), meaning an \(\alpha\)-state is assigned a value of \(0.5\), and a \(\beta\)-state \(-0.5\).

The Coulomb vertex

This is the pivotal quantity, CC-aims computes, which is required by CC4S. In the conventional formulation of most correlated quantum-chemistry methods, like MP2 and CC, the evaluation of the two-electron Coulomb integrals

\[ V_{pqrs} = \int d\mathbf{r} d\mathbf{r'} \psi_{p}(\mathbf{r})^{*} \psi_{q}(\mathbf{r'})\frac{1}{|\mathbf{r} - \mathbf{r'}|} \psi_{r}(\mathbf{r})\psi_{s}(\mathbf{r'}) \]

is necessary, \(\psi_{p}\), \(\psi_{q}\), \(\psi_{r}\) and \(\psi_{s}\) being either MO or Bloch states. This tensor grows with the fourth order with system size and constitutes the major memory bottleneck of post-SCF methods. This is particularly true for CC methods where all interactions between unoccupied/virtual and occupied orbials need to be taken into account.

The Coulomb vertex \(\Gamma_{pq}^{G}\) is a rank-3 tensor, which can be defined by its relation to the tensor of two-electron Coulomb integrals

\[ V_{pqrs} = \sum_{G} \mathbf{\Gamma}_{pr}^{G}(\mathbf{\Gamma}_{qs}^{G})^{\dagger}. \]

By virtue of its reduced rank (in comparison to the Coulomb integrals tensor) the Coulomb vertex constitutes a significantly more memory efficient approach of storing the Coulomb integrals, allowing for the calculation of significantly bigger systems.

For the formula to calculate the Coulomb vertex in a plain-waves basis, see the paper by the Grüneis group. In a localized atomic basis framework, the Coulomb vertex can be obtained via

\[ \Gamma_{ik}^{\nu}(\mathbf{k}_{i}, \mathbf{k}_k) = \sum_{\mu}\sum_{\alpha\gamma} c_{i}^{\alpha*}(\mathbf{k}_{i}) c_{k}^{\gamma}(\mathbf{k}_{k})C_{\gamma\alpha}^{\mu*}(\mathbf{k}_{k}, \mathbf{k}_{i}) (V^{\mathbf{k}_{i}-\mathbf{k}_{k}})^{\frac{1}{2}}_{\mu\nu}. \]

Here, \(\Gamma_{ik}^{\nu}(\mathbf{k}_{i}, \mathbf{k}_k)\) denotes the generalized Coulomb vertex for periodic systems, where \(i\) and \(k\) denote the band-index and \(\mathbf{k}_{i}\) and \(\mathbf{k}_{k}\) the k-vector of two HF-/KS-Bloch states. The \(\nu\)-index goes over all RI/auxiliary basis function with momentum \(\mathbf{k}_{i}-\mathbf{k}_{k}\), \(P_{\nu}^{\mathbf{k}_{i}-\mathbf{k}_{k}}(\mathbf{r})\).

Furthermore, in that equation \(\alpha\) and \(\gamma\) denote Bloch-like basis functions, and \(\mu\) (just as \(\nu\)) denote the index of RI basis functions. \(c_{i}^{\alpha*}(\mathbf{k}_{i})\) is the \(\alpha\)-th element of the \(i\)-th HF- or KS-eigenvector.

\(C_{\gamma\alpha}^{\mu}(\mathbf{k}_{k}, \mathbf{k}_{i})\) denotes the RI-expansion coefficient of the \(\mu\)-th RI-function to approximate the product of the \(\gamma\)-th and the \(\alpha\)-th Bloch-like basis function \(\phi_{\gamma,\mathbf{k}_{k}}\phi_{\alpha,\mathbf{k}_{i}}\).

\((V^{\mathbf{k}_{i}-\mathbf{k}_{k}})_{\mu\nu}\) denotes the auxiliary Coulomb matrix element \(\int d\mathbf{r} d\mathbf{r'} \frac{P_{\mu}^{\mathbf{k}_{i}}(\mathbf{r}) P_{\nu}^{\mathbf{k}_{k}}(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}\).