Atomic orbitals from molecular wave functions: the effective minimal basis

Journal of Physical Chemistry

Volumn 100, Issue 15, 1996, Pages 6249-6257

  Mayer, I ^a  

^a CTRL RES INSTITUTE FOR CHEMISTRY   (Hungary)

Author keywords

[No Author keywords available]

Indexed keywords

CALCULATIONS; MOLECULAR DYNAMICS; PHYSICAL CHEMISTRY;

ATOMIC ORBITALS; EFFECTIVE MINIMAL BASIS; INTRAATOMIC COMPONENTS; LOCALIZED MOLECULAR ORBITALS; MOLECULAR WAVE FUNCTIONS;

MOLECULAR STRUCTURE;

EID: 0030123546     PISSN: 00223654     EISSN: None     Source Type: Journal    
DOI: 10.1021/jp952779i     Document Type: Article

References (32)

1. Foster, J. P.; Weinhold, F. J. Am. Chem. Soc. 1980, 702, 7211.
2. Reed, A. E.; Weinstock, R. B.; Weinhold, F. J. Cheni. Ph\s 1985 83, 735.
3. Reed, A. E.; Curtiss, L. A.; Weinhold, F. Cliem. Rev. 1988 SS 899.
4. McWeeny, R. Rev. Mod. Phys. 1960, 32. 335.
5. Mayer, I. Client. Phys. Lett. 1995, 242, 499.
6. Mayer, I. Can. J. Cliem., accepted for publication in the Bader issue.
7. We recall here the well-known fact that any nonsingular transformation among the occupied orbitals leaves the determinant wave function invariant-except, possibly, a physically irrelevant constant factor. So, it is sufficient to require all the orbitals considered to lie entirely in the subspace of the occupied one-electron orbitals and be linearly independent.
8. Magnasco, V.; Perico, A. J. Chem. Phys. 1967, 47, 971.
9. Bader, R. F. W.; Nguyen-Dana, T. T. Adv. Quantum Chem. 1981, 14, 63.
10. ( 10) The author is extremely indebted to Professor Péter Pulay for calling his attention to this fact.
11. We shall recall in this connection that recently somewhat similar "basis free" definitions of the bond order (multiplicity) between two atoms and of the actual valence of an atom in the molecule have also been given. Angyan, J. G.; Loos, M.; Mayer, I. J. Phys. Chem. 1994, 98. 5244.
12. We use the subscripts/superscripts i only to remember that each atom in the molecule should be assigned its own functional, its own localized orbitals etc., so for each molecule one has to deal with several similar problems simultaneously.
13. If matrix F1 is positive rem/definite, then this can be done in the subspace spanned by the eigenvectors with nonzero A// values.
14. Roby, K. R. Mol. Phys. 1974, 27, 81.
15. Pipek, J.; Mezey, P. G. J. Chem. Phys. 1989, 90, 4916.
16. We recall here that Mulliken's gross atomic population is 2S/\ (PS); for a single orbital one has P = ce.
17. Pipek, J. Int. J. Quantum Chem. 1985, 27, 527;
18. Int. J. Quantum Chem. 1989, .76, 487.
19. The term "intraatomic overlap matrix" has a somewhat different meaning for the case of different functional. In case A it simply gives the overlap of the basis orbitals both belonging to atom A (and is zero otherwise); in case C it represents a "full" matrix formed of the overlap between the projections PAX/I of all basis orbitals in the molecule; in cases B and D it is the matrix of "atomic overlap integrals" Sj),. as defined by eqs 27 and 41, respectively.
20. Amos, A. T.; Hall, G. G. Proc. R. Soc. London 1961, 263A, 483.
21. Löwdin, P.-O. J. Appl. Phys. Suppl. 1962, 33, 251.
22. Mayer, I. Adv. Quantum Chem. 1980, 12, 189.
23. For case A it was shown5 that the LMOs yv can be obtained by solving some îA by m\ equations, too (also vide infra). This may be of importance if one considers extended systems in which the number of occupied orbitals n " mA- (Probably this can also be generalized to case D.)
24. Other variants of the equations may produce some empty effective AOs, too, which we shall not consider here.
25. For these hydrogens one can also obtain up to five effective AOs, similarly to the heavy atoms, but, of course, only one is expected to be of significant importance.
26. Löwdin, P.-O. Phys. Rev. 1955, 97, 1474. '
27. Mayer, I. J. Mol. Struct. (Theochem) 1992, 255, 1.
28. A third related mathematical^ object is matrix SPS consisting of the integrals of the density operator pi: (SPS) V = (x/i\P\(r)\'J.i)', a'so see refs 2 and 26.
29. Note, however, that one has to consider the intraatomic block of the matrix PS-which, in general, differs from P'S, and is closely related to the functional D discussed above-when considering problems related to atomic charge and valences (cf. ref 26): for instance it will give AO occupation numbers which (taking them on all atoms) sum up to the number of electrons.
30. Using the notations above, one may write PA = L'PL', S-= L'SL', ef = L'e, and one obtains eq 49 from eq 17 by substituting eq 22, multiplying with L' from left, and taking into account that L'L' = L1.
31. The directional averaging forces the "pre-NAO"s to keep the spherical symmetry of the free atoms, but in an unsymmetrical environment this symmetry is necessarily spoiled out to some extent by the orthogonalization when the final NAOs are formed.
32. An unpolarized two-center bonding orbital formed of the AOs with overlap S, has a projection [2(1 + 5)]1/2/2 on each of the atoms.