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8.6 Pseudopotentials, Forces and Vibrational Frequencies

It is important to be able to optimize geometries using pseudopotentials and for this purpose Q-Chem contains analytical first derivatives of the nuclear potential energy term for pseudopotentials. However, as documented in Section 8.6.2, these capabilities are more limited than those for undifferentiated matrix elements. To avoid this limitation, Q-Chem will switch seamlessly to numerical derivatives of the ECP matrix elements when needed, which are combined with analytical evaluation of the remainder of the force contributions (where available, as documented in Table 9.1.

The pseudopotential package is also integrated with the vibrational analysis package and it is therefore possible to compute the vibrational frequencies (and hence the infrared and Raman spectra) of systems in which some of the atoms may bear pseudopotentials.

Q-Chem cannot calculate analytic second derivatives of the nuclear potential-energy term when ECP’s are used, and must therefore resort to finite difference methods. However, for HF and DFT calculations, it can compute analytic second derivatives for all other terms in the Hamiltonian. The program takes full advantage of this by only computing the potential-energy derivatives numerically, and adding these to the analytically calculated second derivatives of the remaining energy terms.

There is a significant speed advantage associated with this approach as, at each finite-difference step, only the potential-energy term needs to be calculated. This term requires only three-center integrals, which are far fewer in number and much cheaper to evaluate than the four-center, two-electron integrals associated with the electron-electron interaction terms. Readers are referred to Table 9.1 for a full list of the analytic derivative capabilities of Q-Chem.

8.6.1 Example

Example 8.187  Structure and vibrational frequencies of TeO using Hartree-Fock theory and the Stuttgart relativistic large-core ECPs. Note that the vibrational frequency job reads both the optimized structure and the molecular orbitals from the geometry optimization job that precedes it. Note also that only the second derivatives of the potential energy term will be calculated by finite difference, all other terms will be calculated analytically.

$molecule
   0  1
   Te
   O1  Te  r
   O2  Te  r  O1  a

   r = 1.8
   a = 108
$end

$rem
   JOBTYP     opt
   METHOD     hf
   ECP        srlc
$end

@@@

$molecule
   read
$end

$rem
   JOBTYP      freq
   METHOD      hf
   ECP         srlc
   SCF_GUESS   read
$end

8.6.2 A Brief Guide to Q-Chem’s Built-In ECPs

The remainder of this Chapter consists of a brief reference guide to Q-Chem’s built-in ECPs. The ECPs vary in their complexity and their accuracy and the purpose of the guide is to enable the user quickly and easily to decide which ECP to use in a planned calculation.

The following information is provided for each ECP:

• The elements for which the ECP is available in Q-Chem. This is shown on a schematic Periodic Table by shading all the elements that are not supported.

• The literature reference for each element for which the ECP is available in Q-Chem.

• The matching orbital basis set that Q-Chem will use for light (i.e.. non-ECP atoms). For example, if the user requests SRSC pseudopotentials—which are defined only for atoms beyond argon—Q-Chem will use the 6-311G* basis set for all atoms up to Ar.

• The core electrons that are replaced by the ECP. For example, in the LANL2DZ pseudopotential for the Fe atom, the core is [Ne], indicating that the 1, 2 and 2 electrons are removed.

• The maximum spherical harmonic projection operator that is used for each element. This often, but not always, corresponds to the maximum orbital angular momentum of the core electrons that have been replaced by the ECP. For example, in the LANL2DZ pseudopotential for the Fe atom, the maximum projector is of -type.

• The number of valence basis functions of each angular momentum type that are present in the matching orbital basis set. For example, in the matching basis for the LANL2DZ pseudopotential for the Fe atom, there the three shells, three shells and two shells. This basis is therefore almost of triple-split valence quality.

Finally, we note the limitations of the current ECP implementation within Q-Chem:

• Energies can be calculated only for , , and basis functions with projectors. Consequently, Q-Chem cannot perform energy calculations on actinides using SRLC.

• Analytical ECP gradients can be calculated only for , and basis functions with projectors and only for and basis functions with projectors. This limitation does not affect evaluation of forces and frequencies as discussed in Section 8.6.

8.6.3 The HWMB Pseudopotential at a Glance

8.6.4 The LANL2DZ Pseudopotential at a Glance

Note that Q-Chem 4.2.2 and later versions also support LANL2DZ-SV basis, which employs SV basis functions (instead of 6-31G) on H, Li-He elements (like some other quantum chemistry packages).

8.6.5 The SBKJC Pseudopotential at a Glance

8.6.6 The CRENBS Pseudopotential at a Glance

8.6.7 The CRENBL Pseudopotential at a Glance

8.6.8 The SRLC Pseudopotential at a Glance

8.6.9 The SRSC Pseudopotential at a Glance