Different computational methods are required over the range of sizes from the mercury dimer to the solid state. Firstly, becau e the properties of the system change (e.g. the bulk solid is periodic). Additionally, the computational cost increases with the size of the system, but more quickly with some methods than with others. In this chapter the most general methods are briefly reviewed, and results for the simplest mercury cluster, Hg2 , are examined.
The potential energy curve of Hg2 was calculated using a large, uncontracted basis set, with the use of various density functionals as well as Hartree-Fock, M0ller Plesset perturbation, and coupled cluster methods. This gives an overview of the performance of different methods for this system. These results are compared with those of Zn2, Cd2, and also the group 2 dimer Ba2 having the same 682 configuration as Hg.
Zn and Cd are of interest because they are in the same group as Hg, and can be expected to behave similarly. Differences in behaviour can largely be explained by the influence of relativistic effects. Ba is also of interest for comparison as it does not have the 5d-electrons and so may exhibit marked differences attributable to their presence.
24
Chapter
2.Method review and the dimer potential
2 . 1 Quantum mechanics and moleculesThe underlying problem in all of quantum chemistry is that of charged (charge
q)
and massive (massm)
particles interacting according to the Schrodinger equation:Hw
=Ew
(2. 1 )where the Hamiltonian operator
H
is(2.2)
Atomic units are used such that the electron mass
(me)
and charge(qe)
are 1 . Solv ing the Schrodinger equation for this Coulomb Hamiltonian becomes an increasingly difficult problem as the number of electrons increases, and consequently a series of approximations is necessary. The obvious place to start is by removing the nuclear degrees of freedom, and the fundamental step in electronic structure theory is to sep arate the electronic from the nuclear motion. Effectively this amounts to asssuming that in any motion of the nuclei, the electrons follow instantaneously; this is well justified by the difference between electronic and nuclear mass, a factor of1 03
to 105 , and is known as the Born-Oppenheimer approximation. Thus the total Hamil tonian can be written with nuclear coordinates J.l, v and electronic coordinates i, j as the sum of electronic (kinetic energy), electron-nuclear, electron-electron, and pure nuclear (Coulomb) terms:By applying this to the electronic wavefunction
'l/Jel
(2.4) the electronic energy and wavefunction still depend on the nuclear coordinates R, as well as the electronic coordinates r. For any given nuclear coordinates R, the Hamiltonian can then be written purely in terms of r by neglecting the nuc1ear electron and pure nuclear terms. In practice the errors in the Born-Oppenheimer approximation are negligible for any chemical problem where the notion of molecular structure is valid, and it is this
Eel
which is generally referred to as the exact quantum2.2.
Basis sets
25chemical solution for the electronic Schrodinger equation.
2 .2 B asis sets
Electronic orbitals result from the independent particle model and thus dictate the cost of all quantum chemical calculations. A lot of effort can be saved by construct ing an appropriate set of basis functions that describe these electronic orbitals, in a mathematically simple way while preserving their original properties, such as antisym metry through the use of the Slater determinant. From the hydrogenic Schrodinger equation it can be seen that each orbital or single electron wavefunction is always of the form
(2.5)
where
N
is a normalisation constant and the radial function R(r) can only be found exactly in the case of a one-electron atom. The Ylm (e, cP) are the spherical harmonic functions that describe the angular dependence of the orbital. A practical approxi mation which simplifies the evaluation of two-electron integrals is the Gaussian basis function;(2.6) where P (r) may be a simple polynomial function. In practice the computational requirements can be further reduced by forming contracted Gaussian type orbitals (CGTOs) as a linear combination of GTOs,
XCGTO p = C XGTO ap a . a
(2.7)
Then the number of contracted functions will affect the cost of a calculation. An idea of this is given by the commonly used zeta notation, in which single (SZ), double (DZ), and triple (TZ) basis sets contain one, two, and three times the number of atomic orbitals occupied in the neutral atom. While SZ basis sets are almost always too small to be useful, DZ and TZ basis sets are often used, especially with the addition of higher angular momentum (polarisation) functions.
The way that such basis sets are constructed can vary. For example, dual family basis sets may be used in which one set of exponents is used for each even angular momentum number (s, d, 9, ... ) basis functions
(l
= 0, 2, 4, . . . ) and another set for all odd(l
= 1 , 3, 5, ... ) . This is the case for the large (all-electron) DZ and TZ basis sets used for the dimer polarisability calculations in Chapter 3.26
Chapter
2.Method review and the dim er potential
2 . 3 Hartree-Fock
The electronic Hamiltonian for a many-electron system is
if
=L hi
+L i<j §ij
+ha
(2.8) where the sum over i is of the one-electron terms, and the i < j summation is of two electron interaction terms. Theha
zero-electron term is the inter-nuclear repulsion, a constant for fixed nuclear positions, and can be omitted by the use of the Born Oppenheimer approximation. Then(2.9) is the one-electron kinetic energy and interaction with the nuclei labelled by /-L, and
A 1
9ij
= Tij
(2.10)is the two-electron operator, the Coulomb interaction between two particles of unit charge. The energy of this system as described by this Hamiltonian is evaluated as
the expectation value of the Hamiltonian in terms of the one-electron orbitals
<Pi
1E[1/;] =
(1/; lifl 1/;)
=L . (<Pi Ihl <Pi)
+L (1/; 1§11/;)
i<j
(2.11)where the indices have been dropped from the one and two-electron operators as these are implicit by definition. Then
(2.12)