SCF Results for H2O

• Study convergence of energy, structure and harmonic frequencies with basis set.

• Hartree-Fock limit results from aug-cc-pVQZ-g set. Difference between these and experiment is effect of electron correlation.

• Minimal basis set: STO-3G -- (6s 3p) contracted to [2s 1p] for O based on fit of three GTO to one STO. Results poor.

• Small sp sets: 4-31G and 6-31G -- [3s 2p] contraction of (8s 4p) and (10s 4p), respectively, split-valence or valence double zeta (VDZ). Results fair.

• Other DZ or VDZ sets, e.g., those based on (9s 5p). Results similar to VDZ.

• Need polarization functions for improved results: d sets on O and p set on H, etc.

• Even the smallest polarized basis results are now in good agreement with the largest set.

• Need polarization function on all atoms: 6-31G(d,p) set much better than 6-31G(d) (latter has only d on O).

• Conclude that for structures and frequencies, a DZP basis is rather close to the best Hartree-Fock result. Correlation effects cause the remaining errors.

Molecular Properties

• Other properties, like dipole moment, polarizability, IR intensities, NMR coupling-constants, etc.?

• Consider electric properties first. Molecule in a static homogeneous electric field . Classically,

  

  where is the dipole moment, and is the polarizability tensor. E0 is the energy out of the field, E in the field.

• We can also write a Maclaurin expansion

  

  Hence,

  

  and

  

Properties as Energy Derivatives

• Express properties as derivatives of the energy with respect to "applied perturbations". For example, harmonic force constants are second derivatives with respect to displacing the nuclei.

• Can have mixed second derivatives: differentiating once with respect to an applied electric field and once with respect to nuclear displacements gives a dipole derivative, whose square is the IR intensity.

• Differentiating with respect to an applied magnetic field gives magnetic susceptibility, etc. With respect to nuclear magnetic moments: spin-spin coupling constants, etc.

• How do we compute energy derivatives?

First Derivative of the Exact Energy

• Exact wave function , assumed normalized. Differentiating the energy with respect to gives

  

  

  since the wave function is normalized to a constant (one).

• This is the Hellmann-Feynman theorem.

Calculation of Properties

• Since we do not know the exact wave function, we cannot simply use the Hellmann-Feynman formula.

• Must include the wave function derivatives too.

• Can develop formulas for the energy derivatives: analytical differentiation; or use finite difference formulas based on explicitly including the perturbing operator in the Hamiltonian: finite field methods.

• Can see that if the wave function is independent of the perturbation, will vanish. Thus, for the SCF dipole moment, the Hellmann-Feynman formula is adequate (basis set independent of an applied electric field); for forces on the nuclei we need the full derivative formula, since the basis set depends on the nuclear coordinates.

• No analogue of Hellmann-Feynman theorem for higher-order properties (i.e., higher derivatives).

Basis Sets for SCF Properties

• Dipole moment and polarizability for H2O.

• Convergence is much worse than for structure and frequencies.

• Electric field will affect the most easily polarized part of charge density -- outer fringes. Basis set must include diffuse (low-exponent) functions to describe this effect.

• Electric field multiplies wave function by x, y, or z., hence, polarization functions (diffuse ones) are needed.

• Augment basis sets with more diffuse functions, and with polarization functions. Without these, results are useless.

Correlation Treatments

• H2O results with different treatments: full CI comparison (exact result in DZP basis). Coupled-cluster and MCSCF/CI methods give good results near re, latter are also excellent as bonds are stretched.

• General conclusions from full CI: multireference methods ideal but often too expensive, CC methods good bet where Hartree-Fock is reasonable. Size-extensive methods to be preferred to simple CISD. Perturbation theory not quantitatively useful.

• Basis set studies of H2O: correlation effects converge much more slowly with basis set. DZP is only a starting point, need at least f functions for quantitative results.

• Molecular properties; basis set requirements for describing the perturbations complementary to those for correlation -- use same strategies as when augmenting SCF sets.