Computational Chemistry

Solving this equation givesc 1 (1), an improved version ofc 0 (1). We next solve for

electron 2 a one-electron Schr€odinger equation with electron 2 moving in an

average field due to the electrons ofc 1 (1),c 0 (3),...,c 0 (n), continuing to electron

nmoving in a field due toc 1 (1),c 1 (2),...,c 1 (n#1). This completes the first cycle

of calculations and gives

C 1 ¼c 1 ð 1 Þc 1 ð 2 Þc 1 ð 3 Þ...c 1 ðnÞð 5 : 8 Þ

Repetition of the cycle gives

C 2 ¼c 2 ð 1 Þc 2 ð 2 Þc 2 ð 3 Þ...c 2 ðnÞð 5 : 9 Þ

The process is continued forkcycles till we have a wavefunctionckand/or

an energy calculated fromckthat are essentially the same (according to some

reasonable criterion) as the wavefunction and/or energy from the previous cycle.

This happens when the functionsc(1),c(2),...,c(n) are changing so little from

one cycle to the next that the smeared-out electrostatic field used for the electron–

electron potential has (essentially) ceased to change. At this stage the field of cycle

kis essentially the same as that of cyclek#1, i.e. it is “consistent with” this

previous field, and so the Hartree procedure is called theself-consistent-field-

procedure, which is usually abbreviated as theSCFprocedure.

There are two problems with the Hartree product of Eq.5.7. Electrons have a

property called spin, among the consequences of which is that not more than two

electrons can occupy one atomic or molecular orbital (this is one statement of the

Pauli exclusion principle (Section 4.2.6). In the Hartree approach we acknowledge

this only in an ad hoc way, simply by not placing more than two electrons in any of

the component orbitalscthat make up our (approximate) total wavefunctionc.

Another problem comes from the fact that electrons are indistinguishable. If we

have a wavefunction of the coordinates of two or more indistinguishable particles,

then switching the positions of two of the particles, i.e. exchanging their coordi-

nates, must either leave the function unchanged or change its sign. This is because

all physical manifestations of the wavefunction must be unchanged on switching

indistinguishable particles, and these manifestations depend only on itssquare

(more strictly on the square of its absolute value, i.e. onjcj^2 , to allow for the fact

thatcmay be a complex, as distinct from a real, function). This should be clear

from the equations below for a two-particle function:

If Ca¼fðx 1 ;y 1 ;z 1 ;x 2 ;y 2 ;z 2 Þ

and Cb¼fðx 2 ;y 2 ;z 2 ;x 1 ;y 1 ;z 1 Þ

thenjCaj^2 ¼jCbj^2 if and only ifCb¼CaorCb¼#Ca

If switching the coordinates of two of the particles leaves the function

unchanged, it is said to be symmetric with respect to particle exchange, while if

the function changes sign it is said to be antisymmetric with respect to particle

180 5 Ab initio Calculations