Paris-Parr-Pople approximation

The Paris-Parr-Pople approximation ( PPP approximation ) is a generalization of the Hückel approximation and was proposed by JA Pople in 1953 . Since the approximation referred to as neglecting differential overlap used therein was first proposed by R. Pariser and RG Parr , the term PPP approximation is used.

The term neglect of differential overlap denotes the neglect of all two-electron integrals with the exception of the Coulomb integrals.

description

In the context of the PPP approximation, the eigenvalue problem given by the Schrödinger equation is expressed in an atomic orbital basis in matrix notation . The actual PPP approximation is contained in the approximation for the Hamilton matrix called the Pople matrix : ${\ displaystyle \ mathbb {H} _ {m} \ Psi = E \ Psi}$ ${\ displaystyle H {\ vec {x}} = E {\ vec {x}}}$ ${\ displaystyle P}$ ${\ displaystyle H}$

${\ displaystyle P _ {\ mu _ {i}, \ mu _ {i}} = \ alpha _ {\ mu _ {i}} + {\ frac {1} {2}} c _ {\ mu _ {i} } ^ {2} (\ mu _ {i} \ mu _ {i} | \ mu _ {i} \ mu _ {i}) + \ sum _ {\ nu _ {j} \ neq \ mu _ {i }} c _ {\ nu _ {j}} ^ {2} (\ mu _ {i} \ mu _ {i} | \ nu _ {j} \ nu _ {j}) - \ sum _ {\ nu _ {j} \ neq \ mu _ {i}} Z_ {j} (\ nu _ {j} | \ mu _ {i} \ mu _ {i})}$

${\ displaystyle P _ {\ mu _ {i}, \ nu _ {j}} = \ beta _ {\ mu _ {i}, \ nu _ {j}} - {\ frac {1} {2}} c_ {\ mu _ {i}} c _ {\ nu _ {j}} ^ {*} (\ mu _ {i} \ mu _ {i} | \ nu _ {j} \ nu _ {j})}$ For ${\ displaystyle \ mu _ {i} \ neq \ nu _ {j}}$

The relationship between the molecular and atomic orbital basis is given by , with the atomic orbitals , the orbital coefficients and the orthogonal molecular orbitals . ${\ displaystyle \ phi _ {i} = \ sum _ {\ mu _ {i}} c _ {\ mu _ {i}} \ mu _ {i}}$ ${\ displaystyle \ mu _ {i}}$ ${\ displaystyle c _ {\ mu _ {i}}}$ ${\ displaystyle \ phi _ {i}}$

The one-electron integrals and are formulated analogously to the Hückel approximation. That is, the Hückel matrix is given by for ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$ ${\ displaystyle H '_ {\ mu _ {i}, \ mu _ {i}} = \ alpha _ {\ mu _ {i}}}$ ${\ displaystyle H '_ {\ mu _ {i}, \ nu _ {j}} = \ beta _ {\ mu _ {i}, \ nu _ {j}}}$ ${\ displaystyle i \ neq j}$

Differences to the Hückel approximation

In addition to the Hückel matrix, there are other contributions in the PPP matrix :

(a) repulsion by electrons from other atoms,

{\ displaystyle \ sum _ {\ nu _ {j} \ neq \ mu _ {i}} c _ {\ nu _ {j}} ^ {2} (\ mu _ {i} \ mu _ {i} | \ nu _ {j} \ nu _ {j})}

(b) the attraction of the electrons by the other nuclei,

{\ displaystyle - \ sum _ {\ nu _ {j} \ neq \ mu _ {i}} Z_ {j} (\ nu _ {j} | \ mu _ {i} \ mu _ {i})}

(c) repulsion by electrons on the same atom,

{\ displaystyle {\ frac {1} {2}} c _ {\ mu _ {i}} ^ {2} (\ mu _ {i} \ mu _ {i} | \ mu _ {i} \ mu _ { i})}

and

(d) Terms that represent an attraction of electrons from different atoms and that represent an artifact of the MO approximation

{\ displaystyle - {\ frac {1} {2}} c _ {\ mu _ {i}} c _ {\ nu _ {j}} ^ {*} (\ mu _ {i} \ mu _ {i} | \ nu _ {j} \ nu _ {j})}

.

The contributions (a), (b) and (c) occur in the diagonal elements of the PPP matrix, the terms (d) on the off-diagonal elements.

Parisian-Parr-Pople method

The Pariser-Parr-Pople method ( PPP method ) is a semi-empirical quantum chemical calculation method based on the PPP approximation. The PPP method takes some of the integrals occurring in the PPP approximation as adaptable parameters that were determined, among other things, on the basis of experimental UV spectra of π-electron systems.

literature

Kutzelnigg: Introduction to Theoretical Chemistry. Wiley-VCH, Weinheim 2001.

Individual evidence

↑ JA Pople: Electron interaction in unsaturated hydrocarbons . In: Transactions of the Faraday Society . tape 49 , no. 0 , January 1, 1953, p. 1375-1385 , doi : 10.1039 / TF9534901375 .

↑ Rudolph Pariser, Robert G. Parr: A Semi ‐ Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I. In: The Journal of Chemical Physics . tape 21 , no. 3 , March 1, 1953, p. 466-471 , doi : 10.1063 / 1.1698929 . Rudolph Pariser, Robert G. Parr: A Semi ‐ Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. II . In: The Journal of Chemical Physics . tape
21 , no. 5 , May 1, 1953, p. 767-776 , doi : 10.1063 / 1.1699030 .

[1] JA Pople: Electron interaction in unsaturated hydrocarbons . In: Transactions of the Faraday Society . tape 49 , no. 0 , January 1, 1953, p. 1375-1385 , doi : 10.1039 / TF9534901375 .