# 4.9.5 Non-Hermitian SCF with complex basis functions

(May 16, 2021)

Metastable electronic states can be characterized by a complex Siegert energy,

 $E=E_{r}-i\Gamma/2,$ (4.66)

where the width, $\Gamma$, is proportional to the inverse lifetime of the state: $\Gamma=\hbar/\tau$. Complex coordinate methods aim to compute this complex energy as an eigenvalue of an effective non-Hermitian Hamiltonian. One such method is the method of complex basis functions (CBFs) where a basis of Gaussians with complex exponents is used in conjunction with a symmetric (not complex-conjugated) inner product to effictively produce a finite-basis representation of a non-Hermitian operators. ,,, In cases, such as temporary anions, where the decay channel is of 1-electron character, a mean-field theory can provide approximate Siegert energies for a many-electron system.

The simplest such approximation is the static-exchange approximation. In this approximation the Siegert energies of an ($N+1$)-electron state are computed by diagonalizing a Fock operator computed from the density of an $N$-electron state. This approximation neglects orbital relaxation effects which can be included by a non-Hermitian self-consistent-field (NH-SCF) procedure. , In practice the NH-SCF energy functional is the same as the Holomorphic Hartree-Fock energy functional (Eq. 4.62), though it is used for a different purpose. Both static-exchange and NH-SCF theories using complex basis functions (CBFs) are available in Q-Chem. Specification of the complex basis set is described in Section 8.7.

COMPLEX_EXPONENTS
Enable a non-Hermitian calculation with CBFs.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Perform a non-Hermitian calculation with CBFs
RECOMMENDATION:
Set to TRUE if a non-Hermitian calculation using CBFs is desired.

COMPLEX_SPIN_STATE
Spin state for non-Hermitian calculation
TYPE:
INTEGER
DEFAULT:
1 Singlet
OPTIONS:
$2S+1$ A state of spin $S$
RECOMMENDATION:
None

COMPLEX_N_ELECTRON
TYPE:
INTEGER
DEFAULT:
0 Perform the non-Hermitian calculation on $N$-electrons
OPTIONS:
$n$ Perform the non-Hermitian calculation on an $N+n$ electron system
RECOMMENDATION:
None

COMPLEX_STATIC_EXCHANGE
Perform a CBF static-exchange calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Perform a static exchange calculation FALSE Do not perform a static exchange calculation
RECOMMENDATION:
Set to TRUE if a static-exchange calculation is desired.

COMPLEX_SCF
Perform a non-Hermitian SCF calculation with CBFs
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not perform an NH-SCF calculation 1 Perform a restricted NH-SCF calculation 2 Perform an unrestricted NH-SCF calculation 3 Perform a restricted, open-shell NH-SCF calculation
RECOMMENDATION:
None

COMPLEX_METSCF
Specify the NH-SCF solver
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 Roothaan iterations 1 DIIS 3 ADIIS 21 Newton-MINRES
RECOMMENDATION:
Use the default (DIIS).

COMPLEX_SCF_GUESS
Specify the NH-SCF guess
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use a guess from a static-exchange calculation 1 Read real-basis MO coefficients 2 Read real-basis density matrix 1000 Read guess from a previous calculation
RECOMMENDATION:
Use a guess from a static exchange calculation. Note that for temporary anions, this requires the specification of COMPLEX_TARGET.

COMPLEX_TARGET
Specify the orbital index to be occupied for a temporary anion
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
$n$ Orbital index (starting at zero) for the additional electron
RECOMMENDATION:
$n$ should always be greater than $N_{\text{occ}}-1$.