Calculation of the Electric Dipole Moment of some Diatomic Molecule Hydrides BY F. GRIMALDI," A. LECOURT * AND C. MOSER -/- Received 6th Septetnber 1968 The electric dipole moments for six diatomic molecule hydrides LiH BH AlH FH CIH and NaH have been calculated using functions built from the molecular Hartree-Fock function plus about 200 singly and doubly substituted configurations. The same procedure proved to be quantitatively accurate for CO. In three of the six molecules the electric dipole moment has been measured and the agreement between calculated and observed moments is good. The electric dipole moment of diatomic molecules has been calculated from molecular Hartree-Fock (MHF) functions for several molecules with an error of only a few per cent.'. However when there is near cancellation between the electronic and nuclear contributions to the dipole moment a MHF can give erroneus results.The molecule carbon monoxide is a particular example of this situation. The electronic contribution given by the MHF function is (at Re calc.) - 12.425 a . ~ . ~ and the nuclear contribution is 12.486 a.u. If the MHF function predicts an electric dipole moment (+0-15 D) which is fortuitously in good agreement with the observed moment (0.12 D) the calculated sign (PO-) is contrary to the " observed " sign This coincidence led Nesbet ' to re-examine carefully the hypotheses which were used to deduce the sign of the dipole moment from a microwave e~periment.~ He showed there was an error in the reasoning of Rosenblum et al. who presumed that a positive g factor would lead to completely unreasonable mass ratios.Nesbet has shown that either sign of the g factor would give internally consistent results for the mass ratios. The final conclusion as to the sign of the dipole moment of CO could still be correct. The doubt raised of the interpretation of the microwave experiment has been resolved by Ozier et aL6 who determined the sign of the g factor (and consequently the sign of the dipole moment) from a beam experiment. The sign of the dipole moment in CO seems to be firmly established as C-O+. For molecules with a closed shell Hartree-Fock ground-state doubly-substituted configurations (dsc) give an important contribution to correlation energy but only a very small contribution to the average value of single electron operators. Singly substituted configurations (ssc) contribute much less than do dsc to the correlation energy but make relatively larger contributions to the mean value of the one electron operators.In general only a few out of all possible ssc make significant contributions. These are the ssc which arise by substituting the highest occupied orbital by the lowest unoccupied orbital.8 For the molecule CO adding the 200 most important dsc to the MHF function changes the value of the dipole moment from the MHF value of +0.15 D for +Ow07 (at R = Re ~ a l c ) . ~ Addition of more dsc would change this value of the dipole (c-0'). 5 6 * Commissariat 1'Energie Atomique 29 rue de la FCderation Paris 15e. t Centre de MCcanique Ondulatoire AppliquCe 23 rue du Maroc Paris 19e. 59 60 D I P O L E MOMENT OF DIATOMIC H Y D R I D E S moment by a negligible amount.By contrast a function which is MHF plus the I38 most important dsc plus all 62 ssc (in the basis set used) gives a dipole moment of -0.19 D which certainly confirms the sign which had been deduced from the microwave experiment. The most important contributions which are given by the ssc are those given by the substitution 1z+mn and amongst these the most important are ln+2n and 1n+3z.' We thought it would be of interest to repeat the configura- tion interaction calculation of the electric dipole moment for six diatomic molecule hydrides viz. LiH BH AIH FH CIH as good MHF functions are available from the work of Cade and Huo." We limit our list to these as our configuration inter- action programme is restricted to the use of a closed shell reference state.RESULTS AND DISCUSSION The method of calculation has been described elsewhere 8 r and so it will be necessary only to give a brief r6sumC here. The normalized configuration interaction function has the form ab d 4o is the Hartree-Fock function &{ and 4 are respectively the dcs and SCC ; a b d are occupied orbitals which are replaced by virtual orbitals a j? and y. The c are the coefficients. In the expectation value of the electronic part (z), it is useful to divide up the various contributions to the change A(Z),~ in the H.F. value. I (non diagonal b ) ] (diagonal terms dsc-dsc) terms dsc-dsc) (non diagonal terms dsc-ssc) +C(c:>"(a I z I I z I 4 1 (diagonal terms ssc-ssc) a a a # b \ (non diagonal terms ssc-ssc) (non diagonal terms 4b0-ssc). (2) The largest term in eqn.(2) is generally the last one which is the matrix element of Z between the HF function and a ssc. The reasons for this are (i) As co is nearly equal to one the element is of the order of C," (second order in the present case). (ii) In fact C may be as large as some of the largest C:f (due to summation over inter- mediate states in perturbation series s) when la) and la) are close in energy. F . GRIMALDI A . LECOURT A N D C . MOSER 61 (iii) The integrals ( a I z I a) may be large when the same condition holds which makes C; large. This is due to the fact that I a) and I a) must be orthogonal even with spatial degeneracy (cf. ref. (8)). The other contributions from ssc are negligible (i) The diagonal terms are small since when the difference ( a I z I a)-(a I z I a) is large the probability I C; I is small and vice versa.(ii) Nondiagonal terms are small as the integrals ( a I z I b) ( a I z I p ) are not large and the product of the C is of fourth order. Much the same holds for the contribution of dsc (the coefficients of all terms are of 2nd order). (i) When the difference ( a I z 1 a)-(a I z I a ) is large in the diagonal terms the coefficients C;l are small and conversely. (ii) In the nondiagonal elements the integrals (a I z I a') and ( a I z I a') are never large. Matrix elements of z between ssc and dsc are small as the coefficient CiC is quite small even if the integral ( b I z I p ) may be large. The basis sets for each molecule are those given by Cade and Hu0.l' The R values are those which correspond to the observed equilibrium distance.We have used a programme written by R. K. Nesbet and R. Stevens to calculate the H F functions and orbitals. A large number of dsc can be constructed. The most important of these are chosen amongst all possible dsc by calculating the 2nd order contribution to the energy of each configuration using the Raleigh-Schroedinger method (the sum of single particle Hartree-Fock Hamiltonians is chosen as the unperturbed Hamiltonian). The number of ssc is much smaller. We can choose the most important of these by selecting the single substitutions 4; for which ( a I z I a) is large compared to ( a I z I a) . I a) and I a) are then close in energy-the highest occupied and lowest unoccupied orbitals of the same symmetry type. We limited the number of ssc only for NaH ClH and AlH. For LiH BH and FH all possible ssc were used.For all molecules there are a total-occupied and unoccupied-of 16 (T and 8 rc orbitals. In table 1 we collect the results for (i) the electronic contribution obtained from respectively (a) the H F function ; (b) HF plus 200 dsc ; and (c) HF plus 200 dsc and ssc. Similar results are given for (ii) the total dipole moment i.e. nuclear minus electronic contributions. It is only necessary to use enough dsc to interact with the ssc and it was convenient to limit the sum of both to 200 configurations. For the three molecules for which the dipole moment has been measured LiH FH and ClH the agreement with experiment is good. The error is about 1 1 and 7 % respectively. As the calculated Re values will probably not coincide with the observed R values the larger error for ClH may not be significant.The change in the electronic part in going from function (a) to (c) is 1 4 and 3 % for LiH FH and ClH respec- tively. No experimental measurements are available as far as we know for BH AlH and NaH. AlH will be a particularly sensitive test of the value of our calculations as the function (c) gives a dipole moment which has a different sign from that given by the HF function. In table 2 we give the percentage of the contribution of 2 JTCoECz(a I z I a) to A(z)elect. and show that these terms are by Far the most important. In table 3 we give evidence that the largest part of the sum in table 2 is given by at most two configurations of the type which we have previously mentioned. As the contributions may have both positive and negative signs it is possible to have the contributions total more than 100 %.BH is an exception which merits special mention. The largest coefficients Cz in this molecule are of the order and are at least 3 times smaller than the smallest coefficients quoted in table 3. These coefficients are smaller than those of LiH as the 62 DIPOLE MOMENT OF DIATOMIC HYDRIDES TABLE 1 .-CALCULATION OF THE DIPOLE MOMENTS OF DIATOMIC MOLECULE HYDRIDES molecule LiH BH AIH FH ClH NaH nuclear part (a.u.) 3.015 2.336 3.1 14 1.7328 2.4087 3.566 a (a.u.) 5.3763 1.6541 3.0478 0.9689 1-9381 6.3054 b (a.u.) 5-3658 1.6660 3.044 0.9864 1-9983 6.2717 electronic contribution c (a.u.) 5.3028 1.6663 3.1175 1.0116 1-9984 6-1228 % change (ato c) 1.3 0.73 2.3 4.3 3.0 2.85 a (a.u.) -2-3613 +0-6819 +0.0662 +0.7639 +O-4706 -2.7393 Debye -6.0005 + 1.7338 +0.1682 + 1.9412 + 1.1958 -6.9611 total b (a.u.) - 2-3508 + 0.6700 + 0.0736 + 0.7464 + 0.4503 + 2.7057 c (a.u.) - 2.2878 + 0.6697 - 0.0035 + 0.721 1 + 0.4103 - 2.5567 Debye -5.8137 + 1-7018 -0.0089 + 1.832 1 -0426 - 6.497 1 experimental (Debye) - 5.882 + 1.8195 + 1-12 % error C - 1.19 a + 3.15 0.7 - 6.5 6.6 + 6.9 (a) Hartree-Fock function (ref.(1)) ; (6) H.F.+200 doubly substituted configuration ; (c) H.F.+200 doubly and singly substituted configurations. L. Wharton L. P. Gold and W. Klemperer J. Chenz. Physics 1960 33 1255. R. Weiss Physic. Rev. 1963 131 659. C. A. Burrus J. Chem. Physics 1959 31 1270. TABLE 2.-RELATIVE CONTRIBUTION TO A(Z)elect OF 22/2CoCC,"(a I Z ] a) (3 molecule LiH BH AIH FH CIH NaH % 80 2-5 120 71 77 78 A1H 0.972 42; 0.0183 1.513 0.075 107 * FH 0.982 44; - 0.010 - 0.780 0.0217 51 4:: - 0.022 - 0.047 0.0029 6 4; 0.001 5 0.549 0.0029 6 ClH 0979 $2; - 0.0069 - 1.327 0-0253 42 42: 0.0078 0.7687 0.0166 26 NaH 0.984 42; 0.0409 - 0.9858 -0.185 101 * * As other contributions niay have different signs one niay have more than 100 %.F. GRIMALDI A . LECOURT AND C . MOSER 63 energy difference between the highest occupied and lowest unoccupied orbital is 0.7 a.u. in BH as compared to 0.32 in LiH. The value of A ( Z ) ~ ~ ~ ~ . is small and princi- pally comes from the diagonal dss-dss terms. Our conclusion is that calculation of the important second order contributions to the dipole moment will likely make it possible to produce quantitative estimates of the dipole moment of diatomic molecules even when there is near cancellation between the nuclear and electronic parts. P. E. Cade and W. H. Huo J. Chem. Physics 1966,45 1063. S. L. Kahalas and R. K. Nesbet J. Chem. Physics 39 1963 529. W. H. Huo J. Chem. Physics 1965 43 624. C. A. Burrus J. Chem. Physics 1959 31 1270. B. Rosenblum A. H. Nethercott and C . H. Townes Physic Rev. 1958 109 400. I. Ozier P. Yi A. Khosla and N. F. Ranisey J. Chem. Physics 1967 46 1530. R. K. Nesbet J. Chem. Physics 1964 40 3619. F. Grimaldi Adv. Chern. Physics 1968 14 in press. F. Grimaldi A. Lecourt and C. Moser Int. J. Quantum Chem. 1967 IS 153. P. E. Cade and W. H. Huo J. Chem. Physics 1967 47 614 649.