Chapter 5, Harder Questions, Answers
Q5
In an ab initio calculation on H 2 or HHeþ, one kind of interelectronic interaction
does not arise; what is it, and why?
“Pauli repulsion” does not arise, because there are no electrons of the same spin
present. Of course, this is not a repulsion like that between particles of the same
charge, but just a convenient term for the fact that electrons of the same spin tend to
avoid one another (more so than do electrons of opposite spin). Thus the calculation
of the energy of these molecules does not involve theKintegrals.
Chapter 5, Harder Questions, Answers
Q6
Why are basis functions not necessarily the same as atomic orbitals?
Strictly speaking, atomic orbitals are solutions of the Schr€odinger equation for a
one-electron atom (hydrogen, the helium monocation, etc.). They are mathematical
functions,c, of the coordinates of an electron, and for one electron the square ofcis
an electron probability density function. Solving the nonrelativistic Schr€odinger
gives a series of orbitals differing by the values of the parameters (quantum numbers)
n,l, andm(s orbitals, p orbitals, etc.) [1]. These arespatialorbitals; the relativistic
Schr€odinger equation (the Dirac equation) gives rise to the spin quantum number
s¼½(in units ofh/2p) and to spin functionsaandb, which, multiplied by the spatial
orbitals, givespinorbitals [2]. All this applies rigorously only to one-electron atoms
but has been transferred approximately, by analogy, to all other atoms.
For the integrations in ab initio calculations we need the actual mathematical
form of the spatial functions, and the hydrogenlike expressions are Slater functions
[1]. For atomic and some molecular calculations Slater functions have been used
[3]. These vary with distance from where they are centered as exp(-constant.r),
whereris the radius vector of the location of the electron, but for molecular
calculations certain integrals with Slater functions are very time-consuming to
evaluate, and so Gaussian functions, which vary as exp(-constant.r^2 ) are almost
always used; a basis set is almost always a set of (usually linear combinations of)
Gaussian functions [4]. Very importantly, we are under no theoretical restraints
about their precise form (other than that in the exponent the electron coordinate
occurs as exp(-constant.r^2 )). Neither are we limited to how many basis functions we
can place on an atom: for example, conventionally carbon has one 1s atomic orbital,
one 2s, and three 2p. But we can place on a carbon atom an inner and outer 1s basis
function, an inner and outer 2s etc., and we can also add d functions, and even f (and g!)
functions. This freedom allows us to devise basis sets solely with a view to getting
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