4.9.5 Non-Hermitian SCF with complex basis functions‣ 4.9 Unconventional SCF Calculations ‣ Chapter 4 Self-Consistent Field Ground-State Methods ‣ Q-Chem 6.3 User’s Manual

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

E=E_{r}-i\Gamma/2,

(4.69)

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. ⁸⁹⁷ McCurdy C. W., Recigno T. N.
Phys. Rev. A
(1978), 41, pp. 1364. Link ^, ¹³⁹³ White A. F., Head-Gordon M., McCurdy C. W.
J. Chem. Phys.
(2015), 142, pp. 054103. Link ^, ¹³⁹⁴ White A. F., McCurdy C. W., Head-Gordon M.
J. Chem. Phys.
(2015), 143, pp. 074103. Link ^, ¹³⁹² White A. F. et al.
J. Chem. Phys.
(2017), 146, pp. 234107. Link 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. ¹³⁹³ White A. F., Head-Gordon M., McCurdy C. W.
J. Chem. Phys.
(2015), 142, pp. 054103. Link This approximation neglects orbital relaxation effects which can be included by a non-Hermitian self-consistent-field (NH-SCF) procedure. ⁸⁹⁷ McCurdy C. W., Recigno T. N.
Phys. Rev. A
(1978), 41, pp. 1364. Link ^, ¹³⁹⁴ White A. F., McCurdy C. W., Head-Gordon M.
J. Chem. Phys.
(2015), 143, pp. 074103. Link In practice the NH-SCF energy functional is the same as the Holomorphic Hartree-Fock energy functional (Eq. 4.65), 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

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

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

COMPLEX_N_ELECTRON
       Add electrons for non-Hermitian calculation.
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

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

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

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

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

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 $N_{\text{occ}}-1$ .

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4.9.5 Non-Hermitian SCF with complex basis functions