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# 4.9.4 Holomorphic Hartree-Fock Theory

(February 4, 2022)

## 4.9.4.1 Theory

To use multiple SCF solutions in NOCI (see Section 7.5), it is essential that all solutions exist across all geometries of interest to prevent discontinuities in the NOCI energies. However, it is well known that symmetry-broken SCF solutions can disappear at certain points along a potential energy surface, for example at the Coulson-Fischer point in H${}_{2}$. The holomorphic Hartree-Fock approach provides a means of analytically continuing solutions across all geometries.474, 143, 141

In holomorphic Hartree-Fock theory, the real Hartree-Fock equations are analytically continued into the complex plane without introducing the complex conjugation of molecular orbital coefficients. Multiple solutions are then identified as the stationary points of the holomorphic energy474

 $\widetilde{E}=\frac{\langle\Psi^{*}|\hat{H}|\Psi\rangle}{\langle\Psi^{*}|\Psi% \rangle},$ (4.64)

where $\hat{H}$ defines the conventional electronic Hamiltonian

 $\hat{H}=V_{\text{nuc}}+\sum_{i}^{N_{\text{e}}}\hat{h}(i)+\sum_{i (4.65)

As a result, the holomorphic Hartree-Fock equations are complex-analytic in the orbital coefficients, satisfying the Cauchy-Riemann conditions, and the number of stationary points is found to be constant across all geometries.141 Real Hartree-Fock states remain stationary points of the holomorphic Hartree-Fock energy, and where real solutions vanish, their holomorphic counterparts continue to exist with complex orbital coefficients.143, 141

Holomorphic Hartree-Fock stationary points can be located using minor modifications to conventional SCF algorithms.143. Most significantly, by removing the complex conjugate of the wave function in Eq. (4.64), the required complex holomorphic density $\mathbf{P}$ and Fock $\mathbf{F}$ matrices become complex-symmetric (cf. Hermitian), satisfying $P^{\mu\nu}=P^{\nu\mu}$ and $F_{\mu\nu}=F_{\nu\mu}$. Moreover, since the complex conjugation must also be removed from the normalisation constraint, the molecular orbital coefficients must form a complex-orthogonal set (cf. unitary), i.e.,

 $\sum_{\mu\nu}C^{\cdot\mu}_{i\cdot}S_{\mu\nu}C^{\nu\cdot}_{\cdot j}=\delta_{ij},$ (4.66)

Finally, the holomorphic Hartree-Fock orbital energies and total energy can in general also become complex, and thus selecting the new occupied orbitals on each SCF cycle using the orbital energies is poorly defined. Instead, a complex-symmetric analogue to the Maximum Overlap Method can be employed (see Section 4.5.12).

Following real solutions past the Coulson-Fischer point into the complex plane can often be difficult due to their coalesence with symmetry-pure solutions on the real axis. However, by scaling the electron-electron interaction using a complex parameter $\lambda$, i.e. introducing the Hamiltonian

 $\hat{H}=V_{\text{nuc}}+\sum_{i}^{N_{\text{e}}}\hat{h}(i)+\lambda\sum_{i (4.67)

it is possible to show that Coulson-Fischer points form isolated exceptional points on the real axis.142 Consequently, following a suitable complex $\lambda$ trajectory allows real solutions to be perturbed off the real axis and followed with ease past the Coulson-Fischer point into their complex holomorphic regimes.142 These perturbed solutions can then be relaxed onto the real axis to identify the holomorphic Hartree-Fock states required for NOCI.

## 4.9.4.2 Job Control

Within Q-Chem, the holomorphic Hartree-Fock approach is implemented in the libnoci package (see Section 7.5.2), accessed using USE_LIBNOCI = TRUE and designed for locating multiple SCF solutions for use in NOCI calculations.

SCF_HOLOMORPHIC

SCF_HOLOMORPHIC
Turn on the use of holomorphic Hartree-Fock orbitals.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Holomorphic Hartree-Fock is turned off TRUE Holomorphic Hartree-Fock is turned on.
RECOMMENDATION:
If TRUE, holomorphic Hartree-Fock complex orbital coefficients will always be used. If FALSE, but COMPLEX = TRUE, complex Hermitian orbitals will be used.

SCF_EESCALE_MAG

SCF_EESCALE_MAG
Control the magnitude of the $\lambda$ electron-electron scaling.
TYPE:
INTEGER
DEFAULT:
$10000$ meaning $1.0000$
OPTIONS:
$abcde$ corresponding to $a.bcde$
RECOMMENDATION:
For holomorphic Hartree-Fock orbitals, only the magnitude of the input is used, while for real Hartree-Fock orbitals, the input sign indicates the sign of $\lambda$.

SCF_EESCALE_ARG

SCF_EESCALE_ARG
Control the phase angle of the complex $\lambda$ electron-electron scaling.
TYPE:
INTEGER
DEFAULT:
$00000$ meaning $0.0000$
OPTIONS:
$abcde$ corresponding to $a.bcde$
RECOMMENDATION:
A complex phase angle of $00500$, meaning $0.0500$, is usually sufficient to follow a solution safely past the Coulson-Fischer point and onto its complex holomorphic counterpart.