Effective principal quantum numbers of valence atomic orbitals

VoIume 12, number

CHEMICAL PHYSICS LETTERS

1

EFFECTIVE

PRINCIPAL

QUANTUM

NUMBERS

L.C. CUSACHS

I December 1971

OF VALENCE

ATOMkORBtTALS

and KS. ALDRICH

Richardson Chemisf?y Laboratories. Tulane University, $ew Orleans, Louistim 70118, USA

Received 11 October 1971

Awlvsti of rati mements of Hartree-Fock atomic orbitals show that their shape nea the radial maximum imp&es an effective principal quantum number that remains small even for heavier elements.

The principal quantum number, n, of a canonical solution to the Hartree-Fock equations for an atom is usually defined by the number of nodes it contains. We call this the nominal principal quantum number of the atomic orbital. Another common use of the term principal quantum number is in the radial part of a Slater-type orbital(ST0) [ 11, of the form

Corrington 121. Mann [3] has computed the moments (2) for his solutions to the Hartree-Fock equations for interesting configurations of the elements. TabIe 1 contains Table 1 Effective principal quantum numbers Atom

A n 1 F1~cxp(-zr),

I

(1)

where A,, I is a normalization factor, and n and z are parameters usually called the principal quantum number and orbital exponent respectively. The moments, or averages of powers of r, are easily evaluated for an STO: (I-) = (n +a>/z,

He Be B

C N 0 F Si s

i

Ge Se BI Sn

(2)

The product of(r) and (I/r) is (1 + 1/2n), independent of z! and this product may be used to define an effective principal quantum number for a more general type of orbital. This identification is useful because it defines the n of the ST0 that would reproduce ,these moments for the more gene& function. The significance of matching these moments is described by Cusachs and

N,ffa)

1s

0.886

2s 2s

4s 5s

1.30 1.22 1.18 1.14 1.12 1.11 L.51 1.42 1.38 1.73 1.67 1.64 1.86

5s

1.81

5s 6s

1.79 1.98 1.94 1.49 1.60

2s 2s 2s 2s 3s 3s 3s 4s

cl

cl/rz~=zz/rl(n -3).

Orbital

4

; Pb PO Ni Pt

:: Sd

Orbital

Nerf

2? 2P 2P 3P 3P 3e 4P 4P 4P 5P

1.50 1.43 1.38 1.34 1.32 1.56 L-46 1.43 1.77 1.69 1.65 1.92

6~ 6~ 4s 6s

i-Z 2:04 197 1.80 2.02

a)Ne f is c&&ted from 6) and (l/r) for the orbit& using the tab & ted moments for accurate Hartree-Fock atomic ozbitals of ref. [ 31. These should not be confused with rhe effective valuesof the principal quantum number for owrhp, WIG+ are determined from Gland @).

:

197

Vblume12,number

1

.’

CHEMICALPHYSICS LETTERS

,.

‘.

1 December.1971

‘asample ofeffective principal quantum numbers de-

cipal quantum numbers of atomic orbitals should be

duckd from his (r) and
valence atomic orbit&. It is remarkable that the effective n values remam so small even for rather heavy elements. The low yalues of the effect&e n as compared with the nominal n go far toward explaining the effectiveness [4 - 61 of the,“OS” or of various generalizations

useful in the efficient selection of suitable basis sets for ab initio calculations as well as for more devious semi-empirical manipulations. 7 :

to be effective

or no interestin thecalculation of overlapintegrals

[2] L.C. Cusachs and J.H. Conington in: Signia molecular orbit&J theory, eds.‘O. Sinano~r! and K. Wiberg (Yale Univ. Press, New &ven, 1970). .[3] J.B. Mann, Los Alamos Scientific Reports LA 3690,369l (1967). [4]R.L. Somorjai, Chem. Phys. Letters 2 (1968) 399; 3 (1969)

for semi-empirical molecular calculations, but is a serious consideration when kinetic and more difficult integrals are actuaUy,evaluated. : Awareness of the consistently small effective prin-

[5] D.D. ShilIady,Chem. Phys. Letters 3 (1969) 17.104. [6] G.S. Chandler, Cbem. Phys. Letters 3 (1969) 311. [7] J.H. Ware and R.G. Parr, Intern. J. Quantum Chem. 1S (1967) 163.

References : Hulthdn orbital studied for helium by weare and Parr [7] . [l[ J.C. Slaikr, Physlkev. 36 (1930) 57. The more complicated functional forms all appear in producing

a function

combining

the

correct, or at least,fiite, value near the nucleus with a low effective n in the region where the orbital function is large. The behavior

198

ne21 the nucleus

is of little

395.