• Build a Fock matrix ˜F(n) somehow using C(n) and all insight into the problem
gathered so far.
The final HF energy is given by EC(C0) according to (4.59) and the final SCF orbitals
Θ0by (4.57).
This scheme still leaves a couple of important questions unanswered, which we will address in the following sections, namely:
• What is a suitable method for choosing the initial guess C(0)?
• What type of basis function is suitable?
• What algorithms are sensible for building the next Fock matrix ˜F(n)?
Furthermore remark 5.1 considers the HF problem to be parametrised in terms of
the occupied coefficients C(n) and solves it by producing a sequence of coefficients
C(1), C(2), . . . , C(n)∈ C until convergence. By the arguments discussed in section 4.4.1
one can alternatively parametrise the HF problem in terms of density matrices D(n).
In this light some SCF algorithms are better understood if one thinks about them as schemes producing a sequence of density matrices D(0), D(1), . . . , D(n)
∈ P instead. As an example see the optimal damping algorithm in section 5.4.3 on page 129. To
distinguish both approaches, the first kinds of algorithms iterating C(n) will be called
coefficient-based SCFschemes whilst the second kind of algorithms iterating D(n)we
will call density-matrix-based SCF algorithms. The identification
D(n)= C(n)C(n)
†
,
which we already presented in (4.71) in section 4.4.1, allows to build the density matrix from the coefficients by a matrix-matrix product and in the reverse direction we can find matching coefficients for each density matrix by a factorisation, e.g. a diagonalisation or
a singular-value decomposition4. This allows — at least theoretically — to convert every
density-matrix-based algorithms into a coefficient-based scheme like remark 5.1 and vice versa. In practice, the factorisation from density matrices to coefficients could become rather costly and might not be always applicable.
5.2
Guess methods
A good guess for an iterative procedure like the SCF is characterised by two things. Firstly, it should already be close to the expected solution. Otherwise one might as well start from a random initial set of coefficients. Secondly, it should be cheap to obtain, at least considerably cheaper than the SCF itself. Otherwise again a totally random guess does just as well.
Notice that in general random guesses have not much application from a practical point of view, but for investigating the stability of an SCF procedure they are really helpful. For example, one could check whether a combination of guess method and SCF algorithm yields a true local minimum or just a stationary point of the HF problem (4.65) by trying a few random guesses and checking the resulting energies.
4
88 CHAPTER 5. NUMERICAL APPROACHES FOR SOLVING HF The next sections present a non-exhaustive list of commonly used guess methods for starting SCF procedures.
5.2.1
Core Hamiltonian guess
Only the Coulomb and exchange matrix terms of the Fock matrix expression (4.76)
FCC†= T + V0+ JCC†+ KCC†
depend on the coefficients C. Furthermore the entries of the kinetic matrix T and the
nuclear attraction matrix V0 are typically larger than the entries of the Coulomb and
exchange matrices. A reasonable approximation, which avoids the SCF procedure as a
whole is therefore to find an initial guess C(0) by diagonalising the core Hamiltonian
T+ V0 and keeping the Nelec lowest eigenvalue solutions.
Since electrons in this model do not repel each other the resulting approximate orbitals are typically too contracted and thus not extremely physical. In my calculations with Coulomb-Sturmian-type basis functions (see section 5.3.6) for example I found core Hamiltonian guesses to often converge to stationary points in the SCF process, which are not minima of the HF problem.
Such issues become worse with large basis sets or larger molecules. An ad-hoc way to fix this is to scale the nuclear attraction matrix by a factor 0 < α ≤ 1 in order to mimic the shielding of the nuclear charge somewhat. Nevertheless this guess method is typically only used if other options are not available. An advantage is, however, that it can always be done.
5.2.2
Guesses by projection
A common procedure in many discretisation approaches is to first obtain a quick and crude solution using only a small basis set and to refine the result later in a larger basis. Ideally as much of the information gained in the crude result is used for starting the large calculation.
In the context of the SCF procedure one would, for example, like to use the final coefficients ˜c0from a calculation in the small basis {χν˜}ν=1,...,N˜ g with only Ngfunctions
to obtain a guess for the more refined calculation in the basis {ϕµ}µ=1,...,Nbwith Nb≥ Ng.
Conceptionally this requires to operate on the coefficients with the transformation matrix
U∈ CNb×Ng consisting of elements
Uµ˜ν= hϕµ|χν˜i1.
Since Uµ˜ν is not necessarily unitary, the simple matrix-matrix product Uc0will in general
not be orthonormal with respect to the overlap matrix of the new basis S ∈ RNb×Nb and
is thus not directly usable. Instead, a more involved treatment is required, taking the
overlap matrices both in the old and the new basis into account. If we use ˜s ∈ RNg×Ng
to denote the overlap matrix in the new basis, a properly orthonormalised guess would be [129]
5.2. GUESS METHODS 89 where the matrix
N= ˜c†0U†S−1Uc0 (5.2)
takes care of proper normalisation. The computation (and diagonalisation) of N can
be avoided if other techniques for orthogonalising the Norb column vectors S−1Uc0are
used, like a Gram-Schmidt procedure or a singular-value decomposition. A modification of this procedure would be to alternatively build
f = Uc0Ec˜ †0U†, (5.3)
where (assuming Ng≥ Norb)
˜
E= diag (˜ε1, ˜ε2, . . . ˜εNorb)
are the orbital energies obtained from the SCF in the old basis. Diagonalisation of this
matrix with respect to S yields a set of initial guess coefficients C(0) as the eigenvectors
and the modified orbital energies as the eigenvalues. For example the quantum-chemistry program ORCA uses the latter approach by default [130].
5.2.3
Extended Hückel guess
The extended Hückel (EH) procedure for obtaining estimates of molecular orbitals was developed in the 1960s by Hoffmann [131] based on the extended Hückel Hamiltonian matrix defined in the earlier work by Wolfsberg and Helmholz [132]. Sometimes this procedure is called Generalised Wolfsberg-Helmholz procedure for this reason as well.
The idea is here to start from a minimal set of orbitals {φi}i=1,...,Ntrial, originally
exponential-type orbitals, and build the model Hamiltonian
HijEH= 1 2KS EH ij HiiEH+ HjjEH ,
from the EH overlap matrix SEH with elements
SijEH= Z
R3
φi(r)φj(r) dr,
an empirical parameter K typically set to 1.75 and the diagonal elements HEH
ii , which
should be a rough estimate for the trial orbital energies φi. For this a range of methodo-
logies are employed in practice, including the diagonal elements of the core Hamiltonian matrix of the trial basis, experimental atomic ionisation energies [133] or even the results
from a cheap SCF procedure [130]. The obtained matrix HEH
ij is diagonalised with
respect to the EH overlap matrix SHF yielding trial coefficients CHF. Following the
procedures of the previous section 5.2.2 one may project these onto the basis set of the
problem of interest and use them as an initial guess C(0) for the SCF procedure.
Despite its age the EH method is still subject to active research. For example Lee et al. [134] have constructed a scheme combining the extended Hückel method and Slater’s rules [3] by which decent guesses for finite-element-based density-functional theory calculations may be obtained.
90 CHAPTER 5. NUMERICAL APPROACHES FOR SOLVING HF Typically the EH method only works reasonably well for small basis sets and small molecular systems. This drawback is overcome if an approach related to the superposition
of atomic densities is used for obtaining the diagonal elements of HEH. In the quantum-
chemistry package ORCA [130] for example one can use both the atomic orbitals as well as the orbital energies from pre-calculated atomic STO-3G [4] calculations to drive the EH guess: The trial basis set {φi}i=1,...,Ntrial in their approach is just the combination of
all atomic STO-3G orbitals and the diagonal elements HEH
ii the corresponding STO-3G
orbital energies.
5.2.4
Superposition of atomic densities
The idea of the superposition of atomic densities (SAD) [135] is that molecules are to a very large extend just a collection of atoms, such that the molecular electron density can be obtained approximately just by adding the densities of all constituting atoms together. If atom-centred basis functions are used this process is almost trivial. Let us illustrate the procedure by a chemical system made up of M atoms labelled 1, 2, . . . , M. We first perform atomic ROHF calculations on each atom using the same basis set we want to employ for the molecular calculation, but only the basis functions of the atom in question. This yields converged atomic SCF density matrices D1, D2, . . . DM. In the
SAD guess method as described in [135] the trial density matrix ˜D is the sum of all
density matrix Dα
1, D
β
1, Dα2 after they have been projected from the atomic basis onto
the basis used for the molecular calculation. If we compose the basis of the molecular system in the usual manner, i.e. by pasting together all basis functions a basis set defines
for each atom, in the order atom by atom, ˜Dwould be block-diagonal
˜ D= Dα1 + Dβ1 0 · · · 0 0 Dα2 + Dβ2 · · · 0 .. . ... ... ... 0 0 · · · Dα M+ D β M .
Replicating ˜Dtwice on the α and the β block we can construct the trial density matrix
Dt= ˜D 0
0 D˜
and with it a trial Fock matrix Ft= F[Dt]. A diagonalisation
FtC(0)= C(0)Et
finally yields the initial coefficients C(0) along with some trial energies along the diagonal
matrix Et.
A few remarks about the SAD guess method:
• This whole procedure costs roughly as much as a single SCF step plus the time needed for the atomic calculation, which is typically negligible.
• Furthermore many quantum-chemistry programs store precomputed atomic dens- ities D1, D2, . . . for their supported basis functions and all relevant elements of
the periodic table, such that the cost of the SAD guess procedure is typically even lower in practice.
5.3. BASIS FUNCTION TYPES 91