# Introduction to Computational Chemistry - Jensen Fr.

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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

has

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ELECTRONIC STRUCTURE METHODS

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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