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Szabo A., Ostlund N.S. Modern Quantum Chemistry.. Intro to Advanced Electronic Structure Theory (Dover, 1996)(T)(K)(600dpi)(481s)_Ch_.djvu |
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equation has a solution only if the internuclear distance and basis functions,
and hence the molecular integrals hi h h 22 , etc., are such that" lies between
zero and one, i.e., 0 < " < t...
227
128 MODERN QUANTUM CHEMISTRY
function is not a pure singlet as one would like it to be...
The restrictions on the sum-
mation indices (e.g., a < b, r < s, etc.) insure that a given excited determinant
is included in the sum only once...
When 1'110> is a reasonable approximation to the exact ground
state wave function ICI>o>, the coefficient Co in the CI expansion (see Eq...
This set of coupled eCtuations is extremely
large if all possible excitations are included...
However, the STO-3G
potential energy curve is obtained by subtracting the energy of two isolated
H atoms from the energy ofH 2 and thus need not be an upper bound to the
exact curve...
In this basis, there are 4120 symmetry- and spin-adapted
configurations involving single and double excitations...
A particularly
good test of any procedure for calculating ionization potentials is N 2 since
for this molecule Koopmans' theorem predicts the incorrect ordering of the
first two ionization potentials...
Note that the diagonal element
of the continuous representation of the one-ptatrix is the density of electrons
I'(X h Xl) = P(XI)
(4.36)
Since y(x h X'l) is a function of two variables, it can be expanded in the
orthonormal basis of Hartree- F ock spin orbitals {Xi} as
y(x h X'l) = L Xi(X I )YijX1(x'l) (4.37)
ij
where
'II) = f dX 1 dX'l Xf(Xl)'1( h x )X)(X'l) (4.38)
The matrix y formed from the elements {Yij} is a discrete representation of
the one-matrix in the orthonormal basis {Xi}...
The general equations, which must be
solved to obtain the MCSCF wave function are considerably more com-
plicated than Roothaan's restricted Hartree-Fock equations...
If the same set of basis functions
are used to expand the orthogonal MCSCF orbitals (t/J A and t/JB) and the
nonorthogonal GVB orbitals (u and v), it can be shown (see Exercis{ 4.9)
that the simple two configuration MCSCF wave function of Eq...
(4.23» is
1 Ecorr(DCI) = 1 Ec orr(ex act) = - ( 2 + Kt2)1/2 (4.61)
The deficiency of doubly excited CI, which we discussed at the beginning of
this section, is illustrated in these last two equations...
Are there approximation methods for the correlation energy which are
rigorously size consistent? In fact, there are a number of such schemes and
they constitute the subject matter of much of the rest of this book...
Moreover, there
is numerical evidence that it leads to an improvement over DCI for relatively
small molecules...
Because the energy of a macroscopic system is an extensive thermodynamic
property, it must be proportional to the number of particles; thus DCI is
not a satisfactory procedure for treating large systems...
In particular, the relationship
between different many-electron theories will become clear...
The energy of this wave
function, E ab , is just the sum of the HF energy and the pair correlation
energy ,
Eab = ('P.oIJrI'P 0> + eab = Eo + e ab
(5.5)
Thus the energy of this wave function is below the HF energy by the pair
correlation energy...
Since the total correlation energy using Epstein-Nesbet
pairs, with large basis sets, overestimates the correlation energy even more
than pair theory, it is only occasionally discussed in this book...
In the
case of degeneracies, however, an arbitrary mixing of degenerate orbitals
can occur in the SCF calculation...
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