Electronic wave functions IX. Calculations for the three lowest states of the beryllium atom F. By S. Boys Theoretical Chemistry Department, University of Cambridge {Communicated by Sir John Lennard-Jones, —Received 18 November 1952) The investigation described here has provided wave functions for the lowest S1, P1 and P3 states of the beryllium atom which are more accurate according to the energy test than any wave functions reported previously for these states. The first is an improvement on a previous result by the author, and the other two on results by Hartree & Hartree. The calculations were performed by the poly-detor variational method using exponential functions, and they provided an opportunity to test particular procedures for the choice of elementary functions and detors. The wave functions will be valuable for the calculation of atomic pro perties, but a simple examination is sufficient to show interesting characteristics such as the occurrence of various types of electronic positional correlation. In the calculation various theorems and formulas were established, and these will be useful for further calculations on the same states and also for calculations on many more complicated atomic states.
Electronic wave functions. IX 137 functions themselves can be calculated. Here a superficial examination of the final wave functions makes it possible to see various important characteristics. The occurrence of various types of electronic positional correlation will be shown and discussed in connexion with the Hartree results. The present wave functions make it possible to analyze these factors in some detail.
138 S. F. Boys Notations 8Xi82tpv .,sA,sB,pA,., will be used to denote such elementary sets where the first letter signifies the L value by the spectroscopic code and — Pairs such as sA denote single sets and are used for particular sets of numerical functions, What has been called the serial convention will be used, so that a function known to depend on t4 variables but written without these variables, will be understood to have the arguments tx, t2, tz, reading from left to right.
Electronic wave functions. 139 group. There are a certain number of basic coefficients which must be found by the general theory developed previously, but there are a few times as many coefficients which can be derived from these basic coefficients by simple relations. Here we shall be mainly concerned with the justification of a few such derivation rules. The derivation of the basic coefficients and the application of these rules to deter mine all the Hfg elements required for the present calculations will then be relatively simple.
140 S. F. Boys be noted that the form of the relation implies that it is the primary invariant which is concerned. Then it follows that J2 = (5fS 1siX6L8o)| H | slS1s1 = — (7i(inv2) + S {GrK-2Gxn | xraf\K. (14) rK Proof. It is convenient to let VLS denote the operation R(t,t') .6LS, since this giV68 (F“ || AB) = -ZV%*B„zrx,6™, (15) rs and to use p for any elementary function of X or Y. It then follows by means of the U expansion and insertion of the numerical values of the coefficients given in part VII that ( VLS|| .sfS^S2^, = - \{VLS 1 s^& s^a), slS's^co) + (V3/2) || (16) The removal of the o) operators by theorem 15, part III, and the insertion of the numerical values of the appropriate elementary V coefficients then gives (Fos || s2S1 <§2S2a>, siS1 sxS2ct>) == (~S[S,0] + 8[S,l])s1s2d0S. (17) A simple examination of the V coefficients of the pairs sv and 5351S154S2&>, •s3s2S16’4S2w shows that those of the first pair minus twice those of the second pair give just the above final V expansion. But the s-p interactions of /2 are deter mined completely by this expansion so that /2 = {sxX0lsoj | H | s2 Y0lsoj) -2(szs181s4XdLSG) | H | J54 YdLSoj) + terms involving only s functions. (18) It is apparent that the invariants in both these terms are the same as the invariant of /2, and by an application of theorems 1 and 3, part VIII, it follows that both invariant coefficients are equal to that of Ix, giving a resultant coefficient of — Cx for /2. An obvious application of theorem 3, part VIII, shows that the s-p exchange terms of the first term of (18) have just the coefficients. The s-p exchange terms of the second term are just those of the id terms which would arise if the secondary Schrodinger invariant is taken in this case, thus giving the — 2CxnrDrK coefficients. On the other hand, simple inspection shows that there are no possible s-s inter action terms which are not included in the invariant and that there are no p-p terms at all, since there is a non-coincidence among the s functions. Thus as asserted in the theorem the terms already found constitute all the terms of the integral.
Electronic wave functions. 141 Proof. Let Q stand for either R(t,t') or H, when it follows by the U expansion with the appropriate numerical values of the coefficients and by the usual removal of the co operators that ( Q] {|sfS1 s2S2co} * sxs2S1 s3S2w) IP) = (-1 (QII V(3/2) («|| (20) The second term obviously gives no contribution to the density kernel, and so this for the pair 5fS152S2w, «s1s2S1<s3S2w is just ( —1/^/2) times that for the pair Hence the s-p terms of both the invariant and the rest must be multiplied by (-1/^2) times the corresponding coefficients of v This establishes the first two terms stated in the theorem, and the only remaining term must be a multiple of [«i<s21 s2s3]°, since no p-p terms are possible and this is the only possible s-s term not occurring in the invariant. The coefficient of this is easily found by noting that the ordinary reduction method gives /3 = (sfS152S2w | H j s152S1<s3S2w) ( | and p-p terms. (21) Substitution by (20) and evaluation by the usual WVQ expansion then gives the coefficient 2as in the theorem.
142 S. F. Boys The procedure which has been used for evaluating the variational elements is really more systematic than is necessary for the present cases. If these were the only atomic states to be considered, there would be no case for the construction of all the previous general theory, and the formulas for the variational elements Hrs could have been deri ved by detailed examination and quoted in full as in part II for BeS1. However, the general methods become of great value for more com plicated atoms, and even here they make it easier to avoid errors. For example, the separation of the inv ariant is not only a theoretical convenience: this is computed separately before adding to the variant terms, since it has just the same value for differently coupled cases such as the P1 and the P3.
Electronic wave functions. IX 143 where \sA | sA] has been used to denote (sA | K — ZV | sA)> and similarly. The usual reduction method shows that the variant terms are equal to those of M 2SX IH j sA2Sx) and {sBpAPha | H | sBpSV'o)), the first of these being zero and the second \\sBpA | pAsB~\x from table 1.
144 S. F. Boys 4. Numerical data and procedures The basis of the calculation was a set of radial functions % of form rn exp ( — ar). A set of orthonormal functions <fis was constructed from these and was used without alteration for all three wave functions. These orthonormal space-spin functions are completely described by the coefficients given in table 4 for the radial functions (f> = sa,sb, .,pa,., satisfying (}) — a)rnexp( — (26) n, a The actual elementary functions consist of the sets <f) — sA,sB, .,pA,., where these pairs of letters are to be regarded as single symbols, and are of the form <fi = <j>Slm/i,where Slm denotes the set of spherical harmonics indicated by the first letter of the symbol according to the spectroscopic notation, and denotes the usual spin functions. In principle these should be made orthogonal, but the minor exception that the pD functions are not quite orthogonal to the pG has been per mitted. This is a debatable matter of trivial convenience which does not affect the results significantly. No simple criteria for the choice of these functions are known and they result from the detailed considerations discussed below.
Electronic wave functions. IX 145 empirically by making crude estimates of the amount by which the new co-detors lower the energy. This is generally estimated by assuming — gives the energy lowering due to Or, a result which is exact in the extreme case of small Hrs for 4= Similar tests would be made for a function chosen on external evidence if anything about the numerical results suggested that a better choice might be made. The (n,a) = (0,4), (0,3), (1,1) functions were used in the present case, since they had been found satisfactory in part II for BeS1. The (1,6), used to form the set denoted hj pC, was chosen, since a preliminary calculation for a helium-like atom has shown a maximum lowering of energy for p functions constructed from (1,3Z/2), where Z denotes the nuclear charge. The radial function (1,1) was used for the first p functions, that is, th epAest, and empirical trials were made with (1,2), (1, |), (2,1), (2, l) for other p functions. These indicated that it was much the most satisfactory to use (2,1) to construct pB and (2,^) for the next function designated pi). The radial functions (2,1) and (2,2) were used for the d functions to give some possi bility of variational adjustment while saving a certain amount of computation by using (2,1) for s,p and d functions.