CALCULATIONS IN QUANTUM CHEMISTRY L COMFUTATION OF THE INTEGRALS r .4-1 Aa(o,a),= Sx'exp(--axld:e and B,(a) = ~ ZaeXp(--0LT) da~ ON A DIGITAL ELECTRONIC COMPUTER Yu. A. Kruglyak and D. R. Whitman .~ar'kov State Univc,lry; Case Institute of Technology. Cleveland. Ohio Translated from Zhurnat Strukturnof KhimIl. VoL 3. No. 5. pp. ~9-593. September-October. 196~ Original article submitted September 16. 1961 properties of the integrals An(O. a) and Bn(C~)often encountered In quantum mechanical calculatlooJ of atomic and molecular struct~es are Indicated. and methods of computing these Integrals are ex- posed. Programs In the International algebraic language ALGOL and block schemes for the computa- tion of An(1. o) and Bn(a) on digital electronic computcrs (DEC) are given; they have been checked by the authors and used in quantum chemical calculations. ^ comp|ete ga'vey of published tables containing numerical values of the lntegr..:~ An( ~1. o) and Bn(c{) Is given. I. Main Properties of the Integrals An(O. a) and Bn(a ) Calculation of the properties of atomic and molecular structures by the methods of quantum mechanics Involves the comg~utation of several basic integrals; the latter. In their turn. may be expressed as ftmctlons of auxiliary Integrals. as a~e Am (~, a) = I z'~ exp 1-- r e and B.(~) = z'exp (-~) dz, --I In q'aamum chemical calculations this kind of integral occurs when a radial wave function of the fo!lowing type Is used: X (r) = X [r" exp (-- ar)l. (1) So. for example, inhFl:ogen-like systems,normalization of the radial part of the wave function yields a polynomial In r of degrees n - 1. namely: R,a(r) = ~ (--5) *-r rV'('r~r'1'('~4t+O x lb-I r ~--1 tz~)'/.r (,~ - k) r (~ + t + z) r {k -- t + 1) Tr) 9 Slate~'s atomic orbitals (SAO) [1.2]. whlch represent a single term (k = n- 1) o, e the expansion (2), have found ex~enslve application. In order to characterize an S^O. It is convenient to Inuoduce the "quantum" number p In con- focmlty with the division of electron gates Into groups, as F:.oposed by Slater ['2]. The values ofp for Slater's groups are a~mbled in Table I, the rormallzed radial pan of an SAO may be expressed in the following ways R,~ (,) = F -v' (2v + t) t2~,' (Z)lq"I'r "-' exp[ -- C~ ) (Z) el w,th C~;"(z) = z-s/n-6 = z~.)Iv.