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Introduction to Computational Chemistry - Jensen Fr.

Jensen Fr. Introduction to Computational Chemistry - John Wiley , 2001. - 444 p.
Download (direct link): introductiontocomputat2001.djvu
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Furthermore, the Fock (in an HF calculation) or Hamilton matrix (in a Cl
calculation) will become block diagonal, as only matrix elements between
functions having the same symmetry can be non-zero. The savings depend on
the specific system, but as a guideline the computational time is reduced
roughly by a factor corresponding to the order of the point group (number
of symmetry operations). Although the large majority of molecules do not
have any symmetry, a sizable portion of the small molecules for which ab
initio electronic structure calculations are possible, are symmetric.
Almost all ab initio programs employ symmetry as a tool for reducing the
computational effort.
3.8.3 Ensuring that the HF Energy is a Minimum
The standard iterative procedure produces a solution where the variation
of the HF energy is stationary with respect to all orbital variations,
i.e. the first derivatives of the energy with respect to the MO
coefficients are zero. To be sure this corresponds to an energy minimum,
the second derivatives should also be calculated.15 This is a matrix the
size of the number of occupied MOs times the number of virtual MOs, and
the eigenvalues of this matrix should all be positive for it to be an
energy minimum. A negative eigenvalue means that it is possible to get to
a lower energy state by "exciting" an electron from an occupied to an
unoccupied orbital, i.e. the solution is unstable. In practice the
stability is rarely checked, it is assumed that the iterative procedure
converged to a minimum. It should be noted that a positive definite
second-order matrix only ensures that the solution is a local minimum,
there may be other minima with lower energies. The second derivative
matrix is identical to that arising in quadratic convergent SCF methods
(Section 3.8.1).
The question of whether the energy is a minimum is closely related to the
concept of wave function stability. If a lower energy RHF solution can be
found, the wave function is said to posses a singlet instability. It is
possible that an RHF type wave function is a minimum in the coefficient
space, but is a saddle point if the constraint of double occupancy of
each MO is relaxed. This indicates that a lower-energy wave function of
the UHF type can be constructed, and is called a triplet instability. It
should be noted that in order to generate such UHF wave functions for a
singlet state, an initial guess of the SCF coefficients must be specified
which has the spatial parts of at least one set of a and /3 MOs
different. There are other types of such instabilities, such as relaxing
the constraint that MOs should be real (allowing complex orbitals), or
the constraint that a MO should only have a single spin function.
Relaxing the latter produces the "general" HF method, where each MO is
written as a spatial part having a spin plus another spatial part having
/3 spin.16 Such wave functions are no longer eigenfunctions of the Sz
operator, and not commonly used.
Another aspect of wave function instability concerns symmetry breaking,
i.e. the wave function has a lower symmetry than the nuclear framework.17
It occurs for example for the allyl radical with an ROHF type wave
function. The nuclear geometry has C2v symmetry, but the Nu symmetric
wave function corresponds to a (first-order) saddle point. The lowest
energy ROHF solution has only Cs symmetry, and corresponds to a localized
double bond and a localized electron (radical). Relaxing the double
occupancy constraint, and allowing the wave function to become UHF, re-
establish the correct Civ symmetry. Such symmetry breaking phenomena
usually indicate that the type of wave function used is not flexible
enough for even a qualitatively correct description.
3.8.4 Initial Guess Orbitals
The quality of the initial guess orbitals influences the number of
iterations necessary for achieving convergence. As each iteration
involves a computational effort proportional to M4, it is of course
desirable to generate as good a guess as possible. In some cases
different start orbitals may also result in convergence to different SCF
solutions, or make the difference between convergence and divergence. One
possible way of generating a set of start orbitals is to diagonalize the
Fock matrix consisting only of the one-electron contributions, the "core"
matrix. This corresponds to initializing the density matrix as a zero
matrix, totally neglecting the electron-electron repulsion in the first
step. This is generally a poor guess, but is available for all types of
basis sets and easily implemented. Essentially all programs therefore
have it as an option.
More sophisticated procedures involve taking the start MO coefficients
from a semi-empirical calculation, such as Extended Huckel Theory (EHT)
or Intermediate Neglect of Differential Overlap (INDO) (Sections 3.12 and
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