Computational Chemistry

C 2 n¼ 1 ffiffiffiffiffiffiffiffiffiffi ð 2 nÞ! p ' c 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ c 2 ð 1 Það 1 Þ c 2 ð 1 Þbð 1 Þ(((cnð 1 Þbð ...

special case of a molecule) spin orbitalscaandcb(cf. the minimization of energy with respect to basis function coefficients in S ...

attraction potential energy terms, and electron–electron repulsion potential energy terms (cf. Fig.5.1). This is actually theele ...

Jis called a coulomb integral; it represents the electrostatic (i.e. coulombic) repulsion between an electron inciand one incj, ...

Kij¼ Z c$ið 1 Þc$jð 2 Þ 1 r 12  cið 2 Þcjð 1 Þdv 1 dv 2 ð 5 : 22 Þ Kis called an exchange integral; mathematically, it arises ...

5.2.3.3 The Variation Theorem (Variation Principle) The energy calculated from Eq.5.14is theexpectation valueof the energy opera ...

wherelijare the Lagrangian multipliers; we don’t know what they are, physically, yet (after all, they are “undetermined”). Diffe ...

and similarly forJ^jandK^j. dSij¼ Z dc$ið 1 Þcjð 1 Þdv 1 þc$ið 1 Þdcjð 1 Þv 1 ð 5 : 31 Þ Using fordH,dJ,dKanddSthe expressions i ...

We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able to use the matrix form of a series of such equatio ...

SubstitutingLfrom Eq.5.40into Eq.5.39: F^c¼#^1 2 PL^0 P#^1 c ð 5 : 41 Þ Multiplying on the left byP#^1 and on the right byPwe ge ...

F^c 1 ð 1 Þ¼e 1 c 1 ð 1 Þ F^c 2 ð 1 Þ¼e 2 c 2 ð 1 Þ F^c 3 ð 1 Þ¼e 3 c 3 ð 1 Þ ... F^cnð 1 Þ¼encnð 1 Þ $ð 5 : 47 Þ These equation ...

(this follows simply from multiplying both sides of a Hartree–Fock equation by ciand integrating, noting thatciis normalized) an ...

optimization we seek a global minimum onahypersurfaceinanenergyversus basis function coefficients space defined byC¼f(basis func ...

5.2.3.6 Basis Functions and the Roothaan–Hall Equations 5.2.3.6.1 Deriving the Roothaan–Hall Equations As they stand, the Hartre ...

s=1 m m basis functions sth basis function c of the sth basis function of ith MO yi = ∑csi fs (^) i = 1, 2, 3, ..., m (m MOs) it ...

numbernof pairs of electrons in the molecule, although only thenoccupied spatial orbitals are used to construct the Slater deter ...

Basis functionf 2 gives Xm s¼ 1 cs 1 F 2 s¼e 1 Xm s¼ 1 cs 1 S 2 s Xm s¼ 1 cs 2 F 2 s¼e 2 Xm s¼ 1 cs 2 S 2 s ... Xm s¼ 1 csmF 2 s ...

can be written as a Slater determinant of the “component” spatial wavefunctionsc (by including spin functions), and in principle ...

equation, we might (eschewing a rigorous deductive approach) suspect that the matrix form is the fairly obvious possibility FC¼S ...

F 11 c 11 þF 12 c 21 þF 13 c 31 þ(((¼eðS 11 c 11 þS 12 c 21 þS 13 c 31 þ(((Þ i.e. Xm s¼ 1 csiFrs¼e Xm s¼ 1 csiSrs ð 5 : 60 Þ But ...