7.10 Coupled-Cluster Excited-State and Open-Shell Methods 7.10.11 Partial Auger Decay Widths from Complex-Variable Calculations 7.10.13 Frozen Natural Orbitals in CC, IP-CC, and SF-CC Calculations

7.10.12 Charge Stabilization for EOM-DIP and Other Methods

(September 1, 2024)

The performance of EOM-DIP deteriorates when the reference state is unstable with respect to electron detachment, ⁷⁰⁰ Kuś T., Krylov A. I.
J. Chem. Phys.
(2011), 135, pp. 084109. Link ^, ⁷⁰¹ Kuś T., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 244109. Link which is usually the case for dianion reference states that are employed to describe neutral diradicals by EOM-DIP. These states are often characterized by occupied Hartree-Fock energy levels having positive (unbound) eigenvalues, corresponding to a wave function that is not normalizable. (These are essentially discretized continuum solutions, represented crudely in a Gaussian basis set. ) Similar problems are encountered by all excited-state methods when dealing with excited states lying above ionization or electron-detachment thresholds.

To remedy this problem, one can employ charge stabilization methods. ⁷⁰⁰ Kuś T., Krylov A. I.
J. Chem. Phys.
(2011), 135, pp. 084109. Link ^, ⁷⁰¹ Kuś T., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 244109. Link This approach, which can be used with any electronic structure method, introduces an additional Coulomb potential to stabilize the wave function. There are three ways to do this:

•

Scaling the nuclear charges, which is accomplished using the $rem variable SCALE_NUCLEAR_CHARGE.
•

User-defined nuclear charges, activated by setting CHARGE_STABILIZE = TRUE and specifying the new charges in a $nuclear_charges input section. The format for the $nuclear_charges input section is the same as that of the $van_der_waals section, which is described in Section 11.2.10.
•

Adding a “charged cage”, i.e., an array of point charges around the molecule, which is activated by setting ADD_CHARGED_CAGE = TRUE. Two types of cages (spherical and dodecahedral) are available; the shape, radius, number of points, and total charge of the cage are set by the user.

In the case of EOM methods, a perturbative estimate of the effect of the external Coulomb potential on the EOM energy will be computed when target state densities are calculated, e.g., when CC_EOM_PROP = TRUE. Charge stabilization techniques can be used with other methods including ground state DFT (in order to describe meta-stable ground states) and TDDFT (to improve the description of auto-ionizing resonances). For methods other than EOM, no special correction is applied and one simply obtains the “ordinary” electronic structure but with modified nuclear charges or in the presence of additional point charges. In such cases, it may be advisable to perform several calculations with differing values of the nuclear charges in order to extrapolate the results to the true atomic numbers. In so doing, only calculations for which stabilization is sufficient to obtain a bound-state wave function should be used in the extrapolation. Calculations that result in unbound occupied levels (SCF eigenvalues $\epsilon_{i}>0$ ) should not be taken seriously, as they represent orthogonalized discretized continuum states and not true bound-state solutions. Results from such solutions will vary strongly with respect to the choice of basis set but in ways that are essentially meaningless.

The following descriptions and examples illustrate all three mechanisms of charge stabilization.

SCALE_NUCLEAR_CHARGE

SCALE_NUCLEAR_CHARGE
       Scale the nuclear charges.
TYPE:
       INTEGER
DEFAULT:
       0 do not scale (use true atomic numbers)
OPTIONS:
       $N$ scale the nuclear charges in a way that adds a charge of $N$ /100 (in a.u.)
RECOMMENDATION:
       For EOM methods a perturbative correction can be added in conjunction with this option (as noted above), but for other electronic structure methods once simply gets a traditional calculation but with modified nuclear charges.

Example 7.75 EOM-DIP excitation energies and properties using a charged cage for stabilization.

$molecule
  -2 1
   C   0.000000     0.000000     0.106788
   H  -0.989216     0.000000    -0.320363
   H   0.989216     0.000000    -0.320363
$end

$rem
   METHOD                     eom-ccsd
   BASIS                      6-311g(d,p)
   SCF_ALGORITHM              diis_gdm
   SCF_CONVERGENCE            8
   CC_T_CONV                  8
   EOM_DAVIDSON_CONVERGENCE   5
   CC_SYMMETRY                false
   DIP_SINGLETS               [1]    ! Compute one EOM-DIP singlet state
   DIP_TRIPLETS               [1]    ! Compute one EOM-DIP triplet state
   CC_EOM_PROP                true   ! Compute excited state properties
   ADD_CHARGED_CAGE           2      ! 1 for dodecahedral, 2 for spherical
   CAGE_RADIUS                225    ! Radius = 2.25 A
   CAGE_CHARGE                500    ! Total Charge = 5 a.u.
   CAGE_POINTS                100    ! Place 100 point charges
   INTEGRAL_SYMMETRY          false ! charged cage may violate point-group symmetry
$end

Example 7.76 Stabilization of SO ${}_{4}^{2-}$ by modified nuclear charges.

$comment
   Charge stabilization of an unbound anion (sulfate) by changing nuclear charge
   for S.  Format for nuclear charges section is same as van_der_waals section.
$end

$molecule
   -2 1
   S     0.0000000000    0.0000000000    0.0000000000
   O     0.8960432838    0.8960432838    0.8960432838
   O    -0.8960432838   -0.8960432838    0.8960432838
   O    -0.8960432838    0.8960432838   -0.8960432838
   O     0.8960432838   -0.8960432838   -0.8960432838
$end

$rem
   METHOD             mp2
   BASIS              aug-cc-pvdz
   CHARGE_STABILIZE   1
$end

$nuclear_charges
   1
   16 16.5
$end

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7.10.12 Charge Stabilization for EOM-DIP and Other Methods