Computational Chemistry

charges, have been excluded from consideration here). A high-level calculation places a total charge (carbon plus hydrogen) – al ...

approaches like the PPP method (Section 6.2.2) are available, but mainly because of the phenomenal success of all-valence-electr ...

middle of a conjugated chain should have differentHiiandHijparameters than ones at the end of the chain. Of course, this approxi ...

A 11 x 1 þA 12 x 2 ¼b 1 A 21 x 1 þA 22 x 2 ¼b 2 Using determinants: x 1 ¼ b 1 A 12 b 2 A 22 D x 2 ¼ A 11 b 1 A 21 b 2 (^) (^) D ...

Equation4.74can be generalized tonbasis functions (cf. the matrix of Eq.4.62): H 11 "ES 11 H 12 "ES 12 +++ H 1 n"ES 1 n H 21 "ES ...

The diagonal terms are alwaysxbut the off-diagonal terms, 1 for adjacent and 0 for nonadjacent orbital pairs, depend on the numb ...

x 10 1 x 1 01 x ¼ 0 (4.80) (compare this with the Fock matrix for the propenyl system). Solving this equation (seeSection 4.3.3) ...

" p 2 bc 11 þbc 21 ¼ 0 ; so c 21 =c 11 ¼ p 2 (Recall thecijnotation;c 11 is the coefficient for atom 1 inc 1 ,c 21 is the coeffi ...

and c 21 ¼ ffiffiffi 2 p c 11 ¼ 1 ffiffiffi 2 p and c 31 ¼c 11 ¼ 1 By substituting into the secular equations4.83theEvalues fo ...

the great majority of cases) topzorbitals (taking the framework plane, i.e. the molecular plane, to be thexyplane). 2.Orbital in ...

atomsiandj, which has the important consequence that theSvalues depend on the geometry of the molecule. SinceSis not taken as a ...

The SHM and EHM basis sets are shown in Fig.4.26. 2.The EHM Fock matrix interactions i,jdo not have just two values (aorb) as in ...

fð 1 sÞ¼ z^31 p ^12 expðÞ"z 1 jjr"R1s (4.94) fð 2 sÞ¼ z^52 96 p ^12 jjr"R2sexp "z 2 jjr"R2s1 2  (4.95) where the parameter ...

(«, not«^0 , as the energy will not depend on manipulation of a given set of basis functions) where the matricesH, C, Sand « wer ...

H^0 ¼C^0 eC (^0) " 1 (4.104) In other words, usingS"1/2we transform the original Fock matrixH, which is not directly diagonaliza ...

coordinates), depending on the program. In practice a virtual molecule would likely be created with an interactive model-buildin ...

[ 56 a] for ignoring internuclear repulsion and using a simple sum of one-electron energies was that when the relative energies ...

and f 2 ðHe1sÞ¼ z^3 He p ! 1 = 2 e"zHjr"RHej (4.109) Reasonable values [ 60 ] arezH¼1.24 Bohr"^1 andzHe¼2.0925 Bohr"^1 , ifris i ...

Orthogonalizing matrix As explained above, we (a) diagonalizeS, (b) calculateD"1/2, then (c) calculate the orthogonalizing matr ...

We now have the energy levels ("25.5 and"5.95 eV), but the eigenvectors ofC^0 must be transformed to give us the coefficients of ...