Modulated Perturbation Theory for Molecular Interactions Part 1.-An Exact Second-order Calculation for the Ground State of Hi BY VALERIO * MICHELE and GIUSEPPEMAGNASCO, BATTEZZATI? FIGARI Istituto di Chimica Industriale dell’universiti, 161 32 Genova, Italy Received 21st March, 1975 A perturbation theory which includes exchange appropriate for calculations of molecular inter- actions over the whole range of internuclear separations from the united atom to the separate atoms is proposed in terms of a modified MS-MA approach. Modulation of Ho by suitable parameters is introduced to a first order to keep the electronic component of the Coulombic perturbation V small at all separations, so giving better convergence of the perturbation expansion. An exact calculation of the interaction energy for the ‘Xistate of Hd shows that the results obtained in second order are capable of accuracy over the whole range of internuclear distances.The separation of the interaction energy into Coulombic and penetration components allows a detailed analysis of the physical nature of the interaction to be made. To obtain the complete potential energy curve for atomic and molecular inter- actions by perturbation theory (p.t.) it is necessary to go beyond the first order to account for the long-range part of the potential and at the same time to take into account naclear and electron exchange which is dominant at small separations. Attempts to include such symmetry restrictions into a higher order pet. have stimulated a great deal of research.l It should be emphasized that such a perturbation expansion is neither unique when truncated in low order lm3nor can it converge rapidly at short distances because of the magnitude of the perturbation. Nonetheless both the physical insight that can be obtained into the nature of the interaction and the fact that improvement over the first approximation is dictated by the functional form of the Perturbation itself render a perturbation approach useful. The aim of this series is to present a perturbation theory, which includes exchange, suitable for giving accurate results in the calculation of molecular interactions over the whole range of internuclear separations R from the united atom (R = 0) to the separate atoms (R= co).This is achieved in ternis of a modified Murrell-Shaw- Muslier-Amos (MS-MA) p.t. 49 based on a inodulation of the unperturbed Hamil- tonian Ho to keep the electronic component of the Coulombic perturbation V small for any internuclear separation. In this way a large part of the interaction is accounted for in first order so giving a better converging expansion, especially at small separations. Although giving a first order wavefunction (w.f.) lacking the correct overall sym- metr~,~.6$ the MS-MA theory was chosen because (i) it gives, in second order, the correct asymptotic behaviour of the Coulombic energy at large separations and almost correct (98 %) asymptotic behaviour of the exchange energy, and (ii) it allows a second-order calculation including exchange to be carried out simply in terms of the first-order polarization function a10ne.~ t Laboratorio di Cibernetica e Biofisica del C.N.R., 16032 Camogli, Italy.$ This can give some trouble in calculating expectation values,l but has no effect on the energy obtained in second order by perturbation theory. 22 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI The theory, which enables a separation of the energy into Coulombic and pene- tration (exchange) components to be has been applied here to an exact second-order calculation of the interaction energy for the ground state (2Z:) of the hydrogen molecular ion H;, for which accurate values for comparison are known over a wide range of internuclear separations. 9 Variational approximations to the second-order energy, the study of excited states, and the extension of the theory to cover heteronuclear diatomic molecules and many-electron atoms and molecules which can be tackled in terms of a double per- turbation expansion l3 will form the object of further papers in this series.METHOD According to the Chipman-Bowman-Hirschfelder generalized p.t. for exchange interactions,l the first few MS-MA perturbation equations are : (ffo-~oMo = 0 (14 (~0-E0)~1+~(~-~1)40 (W= 0 WO-Eo)42 +(y-Ed41 -E2A40 = 0, (1c) where Ho is the unperturbed hamiltonian (an unsymmetric hermitean operator), V = H-Ho the unsymmetric hermitean perturbation, A the antisymmetrizer (a projection operator satisfying A2 = A = At) which commutes with the total hamil- tonian (AH = HA) but does not commute with either Ho or V.The related energy corrections are Eo = mi(A$olHol#'o>, Mz = (60lA40)-~7<+old)o> = 1 (24 El = Mo2(A40pq40) (2b) E2 = Mi(4OIV-Elpl). (2c) E2 can be put in the alternative form 79 l4 where is the first-order polarization functi0n.l. *, Conventional ~.t.~* l5 for H; takes the unperturbed systems to be a ground state hydrogen atom located on nucleus A (see fig. 1) (Sa) the interaction with the other proton at B being taken as perturbation 11v = --f-l'b R' This separation of the hamiltonian is appropriate at large distances but fails in the region near the united atom (u.a.) where it cannot provide a good description of the system. In fact the perturbation series for the electronic energy 16* l7 based on (sa), (5b)and (3a) has a rather poor convergence in the u.a.limit Eo = -5, Ei = -1, E2 = -3. (6) S Atomic units (a.u.) are used throughout this paper: 1 hartree (a.u. of energy) = 27.21 eV = 2635.5 kJ mol-I ; 1 bohr (a.u. of length) = 0.529 17 x lo-* cm. MOLECULAR INTERACTIONS This behaviour can be understood in ternis of the multipolar expamion of the Hamiltonian, namely I FIG.1.-Reference coordinatesfor Hi (z-axis from A to B along R). which shows that an enhanced charge-shift is occurring on nucleus A when the proton at B approaches the hydrogen atom. As a consequence, a perturbation theory with good convergence properties at all internuclear separations R should allow in low order a close "following '' of such charge modifications which may be very large at short range.This can be done most easily by including in Ho from the very beginning the majority of the spherical part of the perturbation through the introduction of an " effective " nuclear charge co which accounts for the charge-shift occurring between the interacting partners. Accordingly, our modulated p.t. will be based on an unperturbed Ho describing a 1s electron in the field of a nuclear charge +co located on A (with eigenfunction & and eigenvalue Eo) COH, = -+v2--, qbo = (c:/n)+exp(-Cora), E~ = -12COY (84ra the perturbation due to the proton at B being Co-1v = ---1 +-. 1 pa rb The first term on the right in (8b)can be interpreted as the intra-atomic component of the perturbation V, and acts so as to screen the spherical part of the remaining interatomic potential.We expect co to change from 1 in the limit of large R (vanish-ingly small intra-atomic perturbation) to 2 in the limit of the united atom (vanishingly small electronic perturbation). Therefore we have as R -+ 0 (He+) Eo + -2, E,"+ 0, while as R + 00 (H+H+) zav = ----32,2-r: +Q(R-4)~2 2~3 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI Since the first order energy (p = coR) Eo+G = 4301 = MXA4oIHel40) P is an upper bound to the true energy eigenvalue, we can use the variation theorem to determine co for each internuclear separation R, obtaining in this way the best starting point for our perturbation expansion.This yields for co 1 +e-P(3 ++Qp -p2++p3)+e-2P(3+6p++p2)+eV3q1+$p +$p2 +3p3)co = 1+e-p(2 +2p -+p2++p3)+e-2P(1 +2p ++p2-+p3-+p") (124 with the asymptotic expansions 331 5 +o(p6)small R (12b)co = 2-+p2+p3-2518p4 +18op and co = 1 +e-q( 1 +$p -+p2)+o(e-2p) large R. (1 24 The modulated first-order equation for 4: becomes (co fixed) to be solved with the condition (40fq5p>= 0. We notice that 47 may be split into a purely spherical part 6:' satisfying whose solution is and a part @" satisfying the equation which has been solved in confocal elliptic coordinates p = (ra4-rb)/R, v = (r,-rb)/R by Dalgarno and Lynn l8 (see also Coulson and Robinson 19* 20). It is lossible to show 21 that 4:' contributes to the polarization (Coulombic) energy ' not to the .I: overall second-order energy.For the gerade state of Hz A = $(1 +P),and from (3a)and (8b)we get P being the operator representing nuclear inversion through the midpoint and SOO= {&$'40). The integrals occurring in (16) can be obtained from ref. (15) simply by scaling R in p and taking account of #;'. Asymptotic expansion (see Appendix) of (16) near the united atom limit (He+) shows that the leading term is now of order R4,t as it should be for a second-order t The conventional MS-MA theory (co = 1) yields the ma. asymptotic expansion EZ = -++ RZ+o(~3). MOLECULAR INTERACTIONS theory satisfying the molecular cusp conditions at the two nuclei,16 and that the term of order R5involves a logarithmic term,17 whereas at large separations we get the correct multipolar expansion of the second-order polarization energy 22 (leading term -2.25R-4).The modulation of Ho introduced in first order to keep the elec- tronic component of the perturbation small at all separations yields in second order therefore a p.t. which is qualitatively correct over the whole range of internuclear separations. CALCULATIONS AND RESULTS Calculations for the 'C: electronic ground state of Hi have been performed on the CII 10070 digital computer of the University using double precision and especially developed numerical routines for the accurate evaluation of the exponential integral 24 The results for a wide range of internuclear separations are collected in tables 1-3.The interaction energy for Hi(2Zl)can be written AE = E-E; = AEo+E1+E,+. .. (174 where E: = -3 is the ground state energy of the isolated hydrogen atom, and AEO = -$(c$-1) (184 1.-Hi(2zi). BESTVALUES OF THE MODULATION PARAMETER AND EXACT MS-MATABLE PERTURBATION ENERGIES up TO SECOND ORDER (a.u.) R co EQ Ei EZ 1.o 1.5 2.0 2.5 3.5 4.0 5 .o 6.0 7.0 8.O 9.0 10.0 12.5 15.0 20.0 3.a 1.537 93 1.361 42 1.238 69 1.153 67 1.094 88 1.054 80 1.028 31 1.002 02 0.995 08 0.995 27 0.996 94 0.998 31 0.999 14 0.999 87 0.999 98 0.999 99 -1.182 62 -0.926 74 4.767 18 4.665 48 4.599 38 4.556 30 4.528 71 4.502 03 4.495 09 4.495 28 4.496 95 -0.498 31 4.499 14 -0.499 87 -0.499 98 4.499 99 0.741 62(+0) 0.359 62(+0) 0.180 68(+0) 0.867 22(- 1) 0.349 38(-1) 0.635 15(-2) -0.861 96(-2) -4.171 74(-1) -0.139 80(-1) 4.875 70(-2) 4.478 27(-2) -4.240 78(- 2) 4.115 08(-2) 4.158 84(-3) 4.195 54(-4) 4.245 17(-6) -0.151 75(-1) -0.188 15(-1) 4.183 79(- 1) 4.162 21(-1) 4.135 33(- 1) 4.108 68(-1) 4.847 51(-2) 4.482 22(- 2) 4.259 64(- 2) 4.138 08(-2) 4.753 63(- 3) 4.432 68(-3) 4.263 90(-3) -0.973 84(-4)-0.454 34(- 4) -0.141 85(-4) the energy due to the radial polarization of the atom by the interacting proton.When c0 # 1 a large part of the interaction can thus be accounted for in the first order of perturbation theory, the multipolar part of the exchange-polarization of the atom remaining as a small effect in second order. The best perturbation expansion occur- ring at any R is given in table 1, and the interaction energies summed to various orders are collected in table 2.The exact second-order MS-MA calculation including exchange and the modulation of the perturbation yields interaction energies which are well within 2 % of the exact values given by Peek.12 When co = 1 we recover exactly the results of Chalasinski,' but at large distances (10 bohr) the modulated results are consistently better. The improvement at short separations is on the other hand rather dramatic (table 1 and fig. 3). V. MAGNASCO, M. BATTEZZATI AND G. FIGARI At very short distances the electronic interaction energy for the gerade "C: state of Hl may be written in terms of the u.a. expansion AE, = E,-E: = AEo+E,"+E2+ . . . 17b) where Eg = -2 is now the energy for the 2Sground state of He+, and AEO = -+(~;-4) (18b) is the energy associated with the decrease of the nuclear charge of the u.a.The first order term Efaccounts for the division of the u.a. nuclear charge producing the TABLE2.-H:('E:). INTERACTION ENERGIES TO VARIOUS ORDERS AND COULOMBIC AND PENETRATION COMPONENTS OF THE INTERACTION UP TO SECOND ORDER (a.u.) R AEE AEo = AE@, AE(2) AEcb AEFD 1.o 0.482 20(- 1) -0.682 62(+0) 0.590 01(-1) 0.438 26(- 1) 0.105 84(+0) -0.620 22(-1) 1.5 -0.823 20(- 1) -0.426 74(+0) -0.671 21(-1) -0.859 36(- 1) -0.224 50(-2) -0.836 91(-1) 2.0 -0.102 63(+0) -0.267 18(+0) -0.865 05(-1) -0.104 88(+0) -0.211 18(- 1) -0.837 67(- 1) 2.5 -0.938 23(-1) -0.165 48(+0) -0.787 57(-1) -0.949 79(- 1) -0.207 29(- 1) -0.742 49(-1) 3.0 -0.775 63(- 1) -0.993 86(- 1) -0.644 48(-1) -0.779 82(-1) -0.163 66(-1) -0.616 15(-1) 3.5 -0.608 55(-1) -0.563 08(-1) -0.499 56( -1) -0.608 25(-1) -0.119 40(- 1) -0.488 84(-1) 4.0 -0.460 85(-1) -0.287 14(- 1) -0.373 34(-1) -0.458 09(- 1) -0.839 54(-2) -0.374 14(-1) 5.0 -0.244 20(- 1) -0.203 07(-1) -0.192 05(-1) -0.240 28(-1) -0.400 48(-2) -0.0022(--1) 6.0 -0.11969(-1) 0.490 02(-2) -0.908 05(- 2) -0.116 77(-1) -0.196 17(-2) -0. 71 52(-2) 7.0 -0.559 40(-2) 0.471 08(-2) -0.404 62( -2) -0.542 71(-2) -0.103 35(-2) -0.439 35(-2) 8.0 -0.257 04( -2) 0.304 85(--2) -0.173 42(-2) -0.248 78(-2) -0.590 20(-3) -0.189 76(-2) 9.0 -0.119 54(-2) 0.168 29( -2) -0.724 85(-3) -0.115 75(-3) -0.361 44(-3) -0.796 09( -3) 10.0 -0.578 73(-3) 0.852 98(-3) -0.297 89( -3) -0.561 79(-3) -0.234 10(-3) -0.327 68(-3) 12.5 -0.130 53-3) 0.128 09(-3) -0.307 58(-4) -0.128 14(-3) -0.942 98(- 4) -0.338 43(-4) 15.0 -0.489 38(-4) 0.165 16(-4) -0.303 87(--5) -0.484 73(-4) -0.451 33(-4) -0.333 96( -5) 20.0 -0.142 59(-4) 0.217 79(-6) -0.273 79(--7) -0.142 12(-4) -0.141 82(-4) -0.300 18(-7) a Referred to separate systems (E: = -0.5 a.u.) and including nuclear repulsion.IJ b AE(n) = AE, + C Ei. c Exact interaction energies as given by Peek. l2 i=l TABLE INTERACTION ENERGIES a (a.U.) NEAR THE UNITED ATOM 3.-Hz(2El). ELECTRONIC (He+) R AE0(l)b AEe(*)c CO A&Cl) d AEe(2)f AE g 0.0 0.5 0.0 2.0 0.0 0.0 0.0 0.1 0.504 53 0.016 04 1.979 93 0.021 79 0.021 72 0.021 75 0.2 0.516 47 0.056 11 1.937 40 0.071 67 0.071 11 0.071 4 0.4 0.554 90 0.172 61 1.832 73 0.201 16 0.197 75 0.199 2 0.5 0.578 65 0.237 63 1.778 88 0.268 24 0.262 74 0.265 0 0.6 0.604 10 0.302 65 1.726 19 0.333 25 0.325 52 0.328 5 0.8 0.657 57 0.426 13 1.626 93 0.453 43 0.441 50 0.445 5 1.o 0.711 63 0.536 15 1.537 93 0.559 00 0.543 82 0.548 2 1.25 0.777 04 0.653 44 1.441 87 0.671 71 0.654 01 0.658 2 1.5 0.838 31 0.750 77 1.361 42 0.766 21 0.747 39 0.751 0 1.75 0.894 74 0.831 86 1.294 43 0.845 86 0.826 93 0.829 8 2.0 0.946 22 0.900 12 1.238 69 0.913 49 0.895 11 0.897 3 a Referred to the united atom (E: = -2 a.u.), b first-order Heitler-London, C second-order MS-MA without modulation, first-order theory with modulated perturbation (this paper), fsecond- order MS-MA theory with modulated perturbation (this paper), gexact values as given by Bates et al.' diatomic molecule,' and contains a substaiitial contribution froin exchange inter- actions (table 4).The overall second-order term is small, suggesting that a first-order p.t. based on the united atom would be adequate at small separations provided ex- change and modulation of the perturbation are properly accounted for. The elec- tronic interaction energies (17b) near the u.a. resulting from our modulated pt. are MOLECULAR INTERACTIONS TABLE4.-Hz ('Z:). SEPARATION OF COULOMBIC AND PENETRATION COMPONENTS OF THE IPJJERACTION~ENERGY(REFERRED TO SEPARATE ATOMS) INTO CONTRIBUTIONS FROM DIFFERE~T ORDER (a.U.) R A,?g Eib Eib EF E,Pn 0.2 0.4 0.6 0.8 1.o 0.123 23(+0)' 0.320 54(+0)" 0.510 13(+0)" 0.676 53(+0)= 0.817 37(+0)" 0.123 40(-1) 0.261 91(-1) 0.143 99(-1) -0.169 go(--)' -0.555 63(-1)' -0.642 34(-1)-0.141 47(+0) -0.174 25(+0) -0.173 17(+0)-0.155 96(+0) -0.639 02(-1)-0.145 57(+0)-0.191 27(+0) -0.206 11(+0) -0.202 8l(+O) 0.636 72(-1) 0.138 07(+0) 0.166 52(+0) 0.161 23(+0) 0.140 78(+0) 1.o 1.5 2.0 2.5 3.0 4.0 6.0 8.0 10.0 12.5 15.0 20.0 -0.682 62( +0)-0.426 74(+0) -0.267 18( +0)-0.165 48(+0)-0.993 86(- 1) -0.287 14(- 1) 0.490 02(-2) 0.304 85( -2)0.852 98(- 3) 0.128 09(-3) 0.165 16(-4) 0.217 79(-6) 0.944 43(+0) 0.526 19(+0) 0.307 93(+0) 0.182 14(+0) 0.105 89(+0) 0.294 57(- 1) -0.488 05(-2)-0.304 37(-2)-0.852 61(-3)-0.128 08(-3)-0.165 16(-4) -0.217 79(-6) -0.155 96(+0) -0.101 70(+0)-0.618 63(-I)-0.373 92(- 1) -0.228 71(-1)-0.913 81(-2)-0.198 14(-2) -0.595 OO(-3)-0.234 47( -3)-0.943 07( -4)-0.451 33(-4) -0.141 82(-4) -0.202 81(+0) -0.166 57(+0)-0.127 25(+0)-0.954 20(- 1) -0.709 53(-1) -0.380 77(- 1) -0.173 89(-2)-0.298 25(-3)-0.307 66(-4)-0.303 88(-5)-0.273 79(-7) -0.910 01(-2) 0.140 78(+0) 0.828 87(- 1) 0.434 84(- 1)0.211 70(-1) 0.933 73(-2) 0.663 09( -3)-0.615 07(-3)-0.158 63(-3)-0.294 25(-4)-0.307 68(-5)-0.300 81(-6) -0.263 95(-8) a Referred to united atom (EZ = -2 a.u.).b Electronic contribution only. R/bohr FIG.2.-Electronic interaction energies referred to the united atom (He+) according to different perturbation theories which include exchange. (1) Exact AEe (full line) ; (2) first-order Heitler- London ; (3) second-order MS-MA without modulation ; (4) first-order u.a.theory ;l6 (5) first-order and (0)second-order MS-MA with modulated perturbation. V. MAGNASCO, M. BATTEZZATI AND G. FIGARI compared in table 3 with the exact values and with those resulting from the MS-MA theory without modulation, and are plotted against R in fig. 2 together with the first-order u.a. results of Byers-Brown and P0wer.l The percentage errors = 100 x (approximate -exact)/exact in the u.a. electronic energies resultkg from the different theories are plotted against R for the u.a. region in fig. 3. The validity of the u.a. modhikited p.t. over the whoIe range of internuclear separations is a-pparent from the figiire.At very small seperations our results seem consistently better than those recently obtained from a modified form of Van Vleck's p.t. for degenerate 26 5r -25-201J~lll~~llltllll~ll,, 0 0.5 '1.0 1.5 2 Rlbohr FIG.3.-Percentage error (see text) in the electronic energy near the u.a. (He+) according to different perturbation theories which include exchange. (l), (2) First- and second-order MS-MA with modu- lated perturbation ;(3) second-order MS-MA without modulation ;(4) first-order u.a, theory.16 Of interest in the description of the physical nature of the interaction is the separa- tion of the interaction energy into Coulombic and penetration (exchange) compo- nents, which can be done following our previous work.8-10 In terms of the permuta- tion operator P we can write : Gb= (40lmbo) (194 w = [I +~401~40~l-1~~401~~-~~~l~o~ U9b) MOLECULAR INTERACTIONS AECbconsists of the quasi-classical Coulombic interaction between nuclei and elec- tron 27*28 (" radial " polarization +Coulombic interaction between static densities +Coulombic interaction between transition densities).AEPnis a symmetry-dependent electronic contribution resulting from the physical identity of the interacting particles 10 1 2Rlbohr 3 4 FIG.4.-Hd(2Cl). Exact second-order MS-MA calculation with modulated perturbation : inter-action energy and its Coulombic and penetration components near the bond region. Top :electronic components of the interaction referred to the united atom (He+) at small internuclear separations.and consists of a first-order term (196) describing " non local " interference effects associated with the interpenetration of static charge distributions ** lo and a second- order term (206) giving the exchange part of the polarization. These definitions are completely general and are valid over the whole range of internuclear separations (only electronic contributions being considered in the u.a. region) and agree with the conventional definition of Coulombic and exchange components 79 15* 29* 30 in the limit of large R where overlap and modulation become negligible. The results of such an analysis for Hz(2Cgf)are given in table 4 and in the last two columns of table 2, and are plotted against R in fig.4. The relative importance of Coulombic and penetration components changes according to the internuclear separation. If we broadly distinguish three regions : V. MAGNASCO, M. BATTEZZATI AND G. FIGARI (a) small R,near the united atom; (b) medium range A,near the bond region; (c) large I? ; th!: calculations show that in region (a) the electronic interaction energy is dominated by the Couloinbic component mainly through its zeroth-order term AEo (right top of fig. 4) ;in region (b)the Coulombic components (including nuclear repulsion) cancel to some extent and the largesz term is the first-order penetration which appears as the main fxtor responsible for the existence of a stable chemical bond at R = 2 bohr (GECbwould account only for about 20 % of the bond energy) ; in region (c) the weak attraction is dominated by the second-order Coulombic energy whose leading term goes as and describes the ‘‘dipole ” polarization of the hydrogen atom by the proton.Note, however, that the range of the first-order penetration is much larger than is usudly believeds. 31 and the neglect of exchange effects in the van der Waals region is expected to a-f€ect seriously the calculated value of the interaction. CONCLUSION In conclusion we make the following remarks. (i) Although higher-order p. t. calculations including exchange are difficult and may be no more accurate than the corresponding variational calculations, the per- turbation approach seems preferable for the reasons already mentioned in the intro- duction and also because different physical effects are easily recognised in diEerent orders of the perturbation expansion.Low order treatments may be sufficiently accurate if a convenient unperturbed hamiltoniaii can be chosen on the basis of physical intuition and mathematical corxnience so as to give a rapidly converging series. (ii) The modulation of the perturbation in first order yields a perturbation expan- sion which shows excellent convergence at any R,this being ensured by the perturba- tion vanishing in the limits R -+ 0 and R -+ m. (iii) The second-order MS-MA theory including exchange does seem appropriate for describing long-range “ physical ”as we!l as short-range “ chemical ”interactions, including the very short-range interactions near the united atom.(iv) An exact second-order calculation attains chemical accuracy (roughly, to wit hiil 1 kcal) over the whole range of internuclear separations. (v) The separation of the interaction energy into Couloinbic and penetration com- ponents gives a physically meaningfill picture of the various factors determining the variation of the interaction with internuclear separation, a fact important in develop- ing accurate semiempirical models of molecular interactions. Points (iii) +(iv) are particularly useful because to continue the calculation through the third order of p.t. would require knowledge of the second-order polarization fum- tion +pZ 32 or the first-order exchange functions 8 clnd col defined in the generalized Chipman-Bowman-Hirschfelder theory.In view of the rather complicated expres- sion resulting for the exact second-order energy this would obviously be very difficult. This point calls also for simpler variational approximations to the second-order energy to allow for practical calculations for any system whose first-order equation (3b)cannot be solved in an exact way. APPENDIX Asymptotic expansions of the interaction energy for H2+(2Z,+)according to the second- order MS-MA theory with modulation of the perturbation. MOLECULAR INTERACTIONS (i) Small R (D = 2R) AE:') = 3D2-4D3+&D4+~05+O(D6) (AW E2 = --'034995D4+[~-$(y+1n 2D)]05 +O(D6) (AW to be compared with the exact expansion l7 AEe = +D2-$D3+&$D4+[#-$(y+1n 2D)]D5+O(D6), (A21 y = 0.57721, Euler's constant.For the leading terms of the Coulombic and penetration components near the u.a. we have : = 402 -2~3AE~ +o(~4) (A34 Eib(e) = 5D3+O(D4) (A3b) E';" = -3~2+%~3+0(04) (A34 Eib = -+D2+3D3+O(D4) (A44 EP,"= $D2-5D3+0(D4) (A4b) AEzb = ~D2-+D3+0(D4), AEP"= 0(D4). (A5) (ii) Large R (p = R) which coincides with the exact long-range multipolar expansion of the second-order golariza- tion energy.22 Leading terms of Coulombic and penetration components of the interaction energy at large separations are : AEo = -e-R(1++R-$R2)+O(e-2R) (A74 Eib = e-R(1+4R-+R2)+O(e-2R) (A7b) which shows that to first order AEo exactly cancels out Efb leaving as the dominant first- order contribution the penetration component (A7c).In second order (A66) already gives the Coulombic component, hence : EP," = -&R e-R+O(e-R), (AW giving for the asymptotic overall components up to second order : 2.25AEC~= --R4 +OW6) (A94 AEPn= -3R e-R+O(e-R). (A9b) While (A9a) is exact, (A9b) gives 98 % of the exact asymptotic value of the exchange energy (-R e-R) given by Herring.33 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI We acknowledge financial support by the Italian National Research Council (C.N.R.). We thank Mrs. Graziella Bitossi for typing the manuscript. D. M. Chipman, J. D. 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