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(May 16, 2021)

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

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

where the width, $\mathrm{\Gamma}$, is proportional to the inverse lifetime of the state: $\mathrm{\Gamma}=\mathrm{\hslash}/\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.
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Phys. Rev. A

(1978),
41,
pp. 1364.
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J. Chem. Phys.

(2015),
142,
pp. 054103.
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J. Chem. Phys.

(2015),
143,
pp. 074103.
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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.
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1170
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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.
^{
748
}
Phys. Rev. A

(1978),
41,
pp. 1364.
Link
^{,}
^{
1171
}
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. 4.62), 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

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

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 ${N}_{\text{occ}}-1$.