If the various k(n) can be found, then this equation can be used to compute the order-byorder energy expansion.
Notice that the first-order energy correction is given in terms of the zeroth-order
(i.e., unperturbed) wavefunction as:
Ek(1) = <k| V | k>,
experimental data or results of ab initio calculations are used to determine the parameters in
terms of which V is expressed. The various semi-empirical methods discussed below differ
in the sophistication used to include electron-electron interactions as
Qualitative Orbital Picture and Semi-Empirical Methods F
Some of the material contained in the early parts of this Appendix appears, in
condensed form, near the end of Chapter 7. For the sake of completeness and clarity of
presentation, this material is r
parameterizing the geometry dependence of , ), the differential destabilization of
antibonding orbitals compared to stabilization of bonding orbitals can not be accounted for.
By parameterizing the off-diagonal Hamiltonian matrix elements in the following
C+ (1s22s22px) => C + (1s22s2px2py) ; EC+
are known, the desired VSIP is given by:
IP2pz = IP C + EC+ - EC .
The EA of the 2p orbital is obtained from the
C(1s22s22px2py) => C -(1s22s22px2py2pz)
energy gap, which means that EA2pz = EAC . Some common IP's
Us,s = -0.5(IPs + EAs) -(ZA-0.5) s,s +1/6 (ZA-1.5) G1(s,s)
for Boron through Fluorine's 2s orbitals; and
Up,p = -0.5(IPp + EAp) -(ZA-0.5) p,p +2/25 (ZA-1.5) F2(p,p) +
Here, F2 and G1 represent the well known Slater-Condon integrals in terms o
experimental data (MINDO, MNDO, CNDO/S) or results of ab initio one-electron
calculations (CNDO, INDO, NDDO) to define their parameters.
The CNDO and CNDO/S methods apply the ZDO approximation to all integrals,
regardless of whether the orbitals are locat
a,a = IP a - EAa .
Alternatively, these one-center coulomb integrals can be computed from first principles
using Slater or Gaussian type orbitals.
ii). The off-diagonal coulomb integrals a,b are commonly approximated either by the
expressions that depend on the atomic-orbital-based density matrix , . This quantity is
computed using the LCAO-MO coefficients cfw_Ci, of the occupied molecular orbitals from
the previous iteration of the
F, Ci = i < | > Ci
equations. In particular,
< | - h2 2/2m| > + ( on center a ) , ,
- < | Zae2/|r-Ra| | > = U ,
as the average value of the atomic Fock operator (i.e., kinetic energy plus attractive colomb
potential to that atom's nucleus plus coulomb and exchange interactions with ot
is made, the general four-orbital two-electron integral given above reduces to its zerodifferential overlap value:
< a b|g| c d> = a,c b,d a,b.
This fundamental approximation allows the two-electron integrals that enter into the
expression for the Fock ma
symmetric representation of the group). In terms of the projectors introduced above in
Sec. IV, of this Appendix we must have
A(S) S a* V
not vanish. Here the subscript A denotes the totally symmetric representation of the
group. The symmetry of the
In the specific case considered here, (E) = 4, (2C3) = 1, and (3v) = 0 (You should
try this.). Notice that the contributions of any doubly occupied nondegenerate orbitals
(e.g., a12, and a22) to these direct product characters (S) are unity because for al
The importance of the characters of the symmetry operations lies in the fact that they do not
depend on the specific basis used to form them. That is, they are invariant to a unitary or
orthorgonal transformation of the objects used to define the matrices
(SN,S 1,S 2,S 3)
(SN,S 1,S 2,S 3)
(SN,S 3,S 1,S 2)
(SN,S 2,S 3,S 1)
(SN,S 1,S 3,S 2)
(SN,S 3,S 2,S 1)
(SN,S 2,S 1,S 3)
Here we are using the active view that a C3 rotation rotates the molecule by 120. The
equivalent passive view is that t
Point Group Symmetry E
It is assumed that the reader has previously learned, in undergraduate inorganic or
physical chemistry classes, how symmetry arises in molecular shapes and structures and
what symmetry elements are (e.g., planes, axes of rotation, c
V = eE . r
is odd under the inversion operation (and hence can not be totally symmetric). This same
analysis, when applied to Ek(2) shows that contributions to the second-order energy of an s
orbital arise only from unperturbed orbitals j that are odd und
examining the symmetries of the state of interest k (this can be an orbital of an atom or
molecule, an electronic state of same, or a vibrational/rotational wavefunction of a
molecule) and of the perturbation V, one can say whether V will have a significa
k(1) = j <j| V | k>/[ Ek0 - Ej0 ] |j> .
When this result is used in the earlier expression for the second-order energy correction,
Ek(2) = j |<j| V | k>|2/[ Ek0 - Ej0 ] .
The terms proportional to 2 are as follows:
lk <l|k(1)> <j| V | l>
<l | V | k> = <l | H | k> - k,l Ek0,
because each such determinant is an eigenfunction of H0.
C. The MPPT Energy Corrections
Given this particular choice of H0, it is possible to apply the general RSPT energy
and wavefunction correction formulas developed
Every group member must have a reciprocal in the group. When any group
member is combined with its reciprocal, the product is the identity element.
The associative law must hold when combining three group members (i.e., (AB)C
must equal A(BC).
This means that these D(4) matrices are really a combination of two separate group
representations (mathematically, it is called a direct sum representation). We say that D(4) is
reducible into a one-dimensional representation D(1) and a three-dimensional