SCF Results for H2O
- Study convergence of energy, structure and harmonic frequencies with
- 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
- 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
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
Calculation of Properties
- Since we do not know the exact wave function, we cannot simply use the
- 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
- 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.,
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.
- 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
- 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
- Molecular properties; basis set requirements for describing the perturbations
complementary to those for correlation -- use same strategies as when
augmenting SCF sets.