The performance of EOM-DIP deteriorates when the reference state is unstable with respect to electron detachment,611, 612 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.460) 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.611, 612, 460 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.460 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 ) 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.460
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:
scale the nuclear charges in a way that adds a charge of /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 7.49 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 SYMMETRY false ! charged cage may violate point-group symmetry 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 $end
Example 7.50 Stabilization of SO 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