Ab initio calculations 171

the smallest closed-shell system in which K integrals arise. A detailed exposition of the significance of the HF integrals is given by Dewar [13]. Note that outside the nucleus the only significant forces in atoms and molecules are electrostatic; there are no vague “quantum-mechanical forces” in chemistry [14]. Chemical reactions involve the shuffling of atomic nuclei under the influence of the electromagnetic force.

5.2.3.3 The variation theorem (variation principle)

The energy calculated from Eq. (5.14) is the expectation value of the energy operator i.e. the expectation value of the Hamiltonian operator. In quantum mechanics an integral of a wavefunction “over” an operator, as E in Eq. (5.14) is an integral of over is the expectation value of that operator. The expectation value is the value (strictly, the quantum-mechanical average value) of the physical quantity represented by the operator. Every “observable”, i.e. every measurable property of a system, has a quantum mechanical operator from which the property could be calculated, at least in principle, by integrating the wavefunction over the operator. The expectation value of the energy operator is the energy of the molecule (or atom). Of course this energy will be the exact, true energy of the molecule only if the wavefunction and the Hamiltonian are exact. The variation theorem states that the energy calculated from Eq. (5.14) must be greater than or equal to the true ground-state energy ofthe molecule.

The theorem [15] (it can be stated more rigorously, specifying that must be timeindependent and must be normalized and well-behaved) is very important in quantum chemistry: it assures us that any ground state (we examine electronic ground states much more frequently than we do excited states) energy we calculate “variationally” (i.e. using Eq. (5.14)) must be greater than or equal to the true energy of the molecule. In practice, any molecular wavefunction we insert into Eq. (5.14) is always only an approximation to the true wavefunction and so the variationally calculated molecular energy will always be greater than the true energy. The HF energy is variational (the method starts with Eq. (5.14)) so the variation theorem gives us at least some indication of the true energy and of how good our wavefunction is: the correct energy always lies below any calculated by the HF method, and the better the wavefunction, the lower the calculated energy. The HF energy actually levels off at a value above the true energy as the HF wavefunction, based on a Slater determinant, is improved; this is discussed in section 5.5, in connection with post-HF methods.

5.2.3.4Minimizing the energy; the HF equations

The HF equations are obtained from Eq. (5.17) by minimizing the energy with respect to the atomic or molecular orbitals The minimization is carried out with the constraint that the orbitals remain orthonormal, for orthonormality was imposed in deriving Eq. (5.17). Minimizing a function subject to a constraint can be done using the method of undetermined Lagrangian multipliers [16]. For orthonormality the overlap integrals S must be constants and at the minimum the energy is constant Thus at any linear combination of E and is constant:

Ab initio calculations 173

Using for dH, dJ, dK and dS the expressions in Eqs (5.26), (5.27), (5.28) and (5.31), Eq. (5.25) becomes

Since the MOs can be varied independently, and the expression on the left side is zero, both parts of Eq. (5.32) (the part shown and the complex conjugate) equal zero. It can be shown that a consequence of

is that

i.e.

Equation (5.34) can be written as:

where is the Fock operator:

We want an eigenvalue equation because (cf. section 4.3.3) we hope to be able to use the matrix form of a series of such equations to invoke matrix diagonalization to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue equation, because it is not of the form Operation on function = k × function, but rather Operation on function = sum of (k × functions). However, by transforming the molecular orbitals

174 Computational Chemistry

to a new set the equation can be put in eigenvalue form (with a caveat, as we shall see). Equation (5.35) represents a system of equations

There are n spatial orbitals since we are considering a system of 2n electrons and each orbital holds two electrons. The 1 in parentheses on each orbital emphasizes that each of these n equations is a one-electron equation, dealing with the same electron (we could have used a 2 or a 3, etc.), i.e. the Fock operator (Eq. 5.36) is a one-electron operator, unlike the general electronic Hamiltonian operator of Eq. (5.15), which is a multi-electron operator (a 2n electron operator for our specific case). The Fock operator acts on a total of n spatial orbitals, the in Eq. (5.35).

The series of equations (5.37) can be written as the single matrix equation (cf. Eq. (4.50))

i.e.

In Eqs (5.37), each equation will be of the form which is what we want, if all the except for i= j (e.g. in the first equation if the only nonzero is ). This will be the case if in Eq. (5.39) L is a diagonal matrix. It can be shown that L is diagonalizable (section 4.3.3), i.e. there exist matrices P, and a diagonal matrix such that

Substituting L from Eqs (5.40) into (5.39):

Multiplying on the left by and on the right by P we get

which, since can be written

Ab initio calculations 175

where

We may as well remove the factor by incorporating it into and we can omit the prime from (we could have started the derivation using primes on the then written for Eq. (5.43)). Equation (5.42) then becomes (anticipating the soon-to-be-apparent fact that the diagonal matrix is an energy-level matrix)

where

Equation (5.44) is the compact form of (cf. Eq. (5.38)). Thus

where the superfluous double subscripts on the have been replaced by single ones. Equations (5.44/5.46) are the matrix form of the system of equations

These Eqs (5.47) are the HF equations (the matrix form is Eqs (5.44) or (5.46)). By analogy with the Schrödinger equation we see that they show that the Fock operator acting on a one-electron wavefunction (an atomic or molecular orbital) generates an energy value times the wavefunction. Thus the Lagrangian multipliers turned out to be (with the the energy values associated with the orbitals Unlike the Schrödinger equation the HF equations are not quite eigenvalue equations (although they are closer to this ideal than is Eq. (5.35)), because in the Fock operator is itself dependent on in a true eigenvalue equation the operator can be written down without reference to the function on which it acts. The significance of the HF equations is discussed in the next section.