Angular Correlation Diagrams for AH,-type molecules BY C. A. COULSON AND A. H. NEILSON* Mathematical Jnstitute, 10 Parks Road, Oxford Received 10th January, 1963 A study is made of the theoretical basis of Walsh’s rules for discussing the equilibrium shapes of polyatomic molecules in ground and excited states. The particular series AH2 is considered, and detailed numerical calculations are made for H20. It is shown that the use of ionization potentials in Walsh’s correlation diagrams may lead to erroneous conclusions, and the importance of the nuclear repulsion energy is emphasized. A scheme of partitioning the total electronic energy in such a way that the total energy is additive with respect to certain defined orbital energies, is described, and illustrated. It is now almost ten years since Walsh 1 published his stimulating series of correlation diagrams for molecules of different symmetries.These diagrams have been widely used for interpreting the change in shape of a molecule when any given electronic excitation takes place. A particular example, with which we shall be much concerned in this paper, is the series of molecules AH2. The water molecule H20 is the best-known member of this series, but in drawing the correlation diagram (fig. 1, reproduced from ref. (1)) it is assumed that by a suitable change of scale all molecules of this kind are included on the one diagram. An example, taken from Walsh’s paper, will serve to show the use of this diagram, and also introduce us to our main problem, which is to understand what is being represented in curves such as those in fig.1 (the curve involving only the 1s orbital of A has been omitted). In the ground state of the radical NH2 there is one electron in the (lbl) orbitaf, and an allowed long-wavelength transition should be described by the formula (lal)2(2al)2(lb2)2(3al)z(lbl), ~ B I t(la1)2(2a1)2(1b2>2(3al)(l61)2, %. This excitation involves the transfer of an electron from the “ steep ” curve (3al) to the almost horizontal curve ( l b l ) . We remove an electron from an orbital where it is trying to reduce the HNH angle and put it in an orbital where it hardly influences the angle. So in the excited 2A1 state the valence angle should be increased. Arguments of this kind are satisfying on account of their simplicity. But before we can make them, we must first know how to plot the correlation diagrams on which the argument depends.This means that we must know what the ordinate represents. In Walsh’s papers, and in most others in which use is made of these diagrams, the ordinate is described as the binding energy of the individual orbital. This is usually interpreted to be the ionization potential. First, the essential idea implied in these curves is that of a one-electron model, in which we can speak unequivocally of the energy of each electron, and consider how this energy varies with valence angle. But it is not satisfactory to use the ionization potential as this energy. This is because we are concerned with the total molecular energy, and this is far from being the same as the sum of all the ionization potentials.Such a sum includes all inter- electron Coulomb and exchange interactions twice, and since these energies vary Some comments are necessary. * present address : Chemistry Department, The University, Glasgow. 7172 CORRELATION DIAGRAMS considerably with angle we may be led into serious error if we count each of them twice. A further point arises-that if we are concerned with variations in valence angle, we must add to the sum of the individual electronic energies the sum of all the nucleus-nucleus Coulomb repulsions. In the usual application of Walsh's diagram no reference is made to this, even though, as we shall see later, this inter- nuclear Coulomb repulsion curve is quite steep. Some of these points are illustrated in our previous example of NH2. Let us consider two valence angles a and fl (acfl).The energy difference between the lower orbital (3al) and the upper orbital (lbl) is less at p than at a. But in going from a to /3 we also change all the other electronic energies and the nuclear repulsions. 90" 1 80° bond angle (a) FIG. 1. I 90" I bond angle (a) Fro. 2. OC Without considering all of these we can say almost nothing about any likely change in valence angle in the excitation process involved. In an extreme case, dealing with an ionization process rather than an excitation, and in which the nuclear repulsions curve fell more steeply than the correlation curve for the electron being removed, it would be possible for a naive use of the correlation diagram to predict a change in valence angle in the opposite direction to the true one.An attempt has recently been made by Schmidtke and Preuss2 to provide a wave-mechanical basis for the curves in fig. 1. They replaced the true complete Hamiltonian by a pseudo-Hamiltonian, in which all the rl2-terms had been deleted, and an effective nuclear charge ZA replaced the true nuclear charge on the central atom A. In this way all Coulomb and exchange terms disappear ; the total energy is just the sum of the separate one-electron energies: but of course the nuclear repulsion energy must be added as an additional factor when discussing possible changes in valence angle due to excitation or ionization. There is, unfortunately,C . A. COULSON AND A. H. NIELSON 73 some doubt as to the appropriate ZA to use (Schmidtke and Preuss favour ZA = 4.5 for HzO, though this value would clearly be most inappropriate for the K-shell electrons of the oxygen atom).But there is a fairly close parallel between the resulting curves (fig. 2) and those originally proposed by Walsh and reproduced in fig. 1. However, our fundamental objections remain with any one-electron model where the Coulomb and exchange energies are neglected, or an attempt is made to com- pensate for them by an appropriate pseudo-Hamiltonian. It therefore seemed desirable to see whether a treatment of this problem could be presented, which in- cluded Coulomb and exchange energies, and at the same time had an additive (or nearly additive) character. Such a treatment will be attempted in what follows.In the process of developing the theory we shall be able to state more explicitly what is really being plotted as ordinate in these correlation diagrams. We shall agree with Schmidtke and Preuss that it is very far from being the ionization potential ; but, unlike them, in some situations we shall be able to specify it more completely. ONE-ELECTRON ENERGIES Consider a molecule with 2n electrons occupying the n molecular orbitals $1, 4 2 , . . &. We suppose these functions to be orthonormal. If these 4i are the lowest lying orbitals then the ground state is represented symbolically by q5:& . . .q5$ The complete Hamiltonian may be written in the form In where H(i) is the core Hamiltonian for electron i, VN is the internuclear Coulomb repulsion and rtj is the distance between electrons i and j.It is well known that the energy of the state considered may be expressed as where Et is the core energy for orbital &? given by Ei = 1 + r ( 1 ) ~ ( 1 ~ i ( 1 ) d z l , and where J and K are the familiar Coulomb and exchange integrals between the molecular orbitals 4i and $ j . In (2), every summation is over all i a n d j in the range 1 ton. If we accept the validity of Koopmans’ theorem, then the ionization potential of the molecular orbitalq5t is -EZ, where is the molecular-orbital energy. It is immediately clear from (2) and (3) that if we make the sum of all ionization potentials, by writing Q = 2CEi (4) then 8‘ differs from the total energy E in two respects-first, by counting the Coulomb and exchange terms twice, and secondly, by omitting the nuclear energy term VN.These two matters show why in drawing correlation diagrams it is not correct to plot ionization potentials. But of course we cannot plot the core energies either. If we want to have some quantity associated with the separate orbitals and which is truly additive, then we74 CORRELATION DIAGRAMS may obtain such a quantity by partitioning the energy * in the form E = 2&+ V,, where ( 5 ) = &(Ei + ~ i ) (7) If we now draw curves showing how each ea varies with the valence angle a, and then include the nuclear energy V', we should find a minimum of E at the equilibrium value of a. Further, each separate curve tells us the influence of the appropriate molecular orbital, in the presence of the other orbitals, in trying to open or close the valence angle.CALCULATIONS In order to show how ei(a) depends on a for a particular molecule, we have used the molecular orbitals for H20 as determined by Ellison and Shull,3 who made Roothaan-type self-consistent 1.c.a.o. calcu1ations.t Each molecular orbital 4i is given in terms of known atomic orbitals, and so all the required Coulomb and exchange integrals may be found. A matrix formulation appeared to be the most convenient, and the whole calculation was programmed for the Oxford University Mercury computer. Fig. 3 shows the variation with bond angle a of each ei(a), and should be com- pared with fig. 4 which shows the variation of the calculated ionization potentials -&&(a). The curves are labelled by the symmetry of the m.o.'s and the la1 curve has been omitted in both cases since the m.0.is composed almost entirely of the oxygen 1s orbital. By comparison of fig. I and fig. 4 it can be seen that the calculated ionization potentials and Walsh's binding energies are very similar, both in relative value and in their variation with bond angle.$ Comparison with the partitioned energies ei, however, shows that ei and E$ have entirely different angular variation for most orbitals? and that they do not even lie in the same order. In fig. 3 the ( 3 4 level has much the greatest angular variation, with a minimum at a = 180". The nuclear repulsion energy V N ( ~ ) also has its minimum at 180". On our present interpretation these are the two factors largely responsible for increasing the bond angle from 90".But if we use the ionization potential diagrams in fig. 4 it is only the (lb2) orbital which tends to open out this angle. The difficulty in using either the Walsh diagram (fig. 1) or the Schmidtke-Preuss diagram (fig. 2) is that, as they stand, they both give no expectation that the equilibrium angle is greater than 90". We have made some simple calculations (see table) using the same numerical values as Schmidtke-Preuss : they indicate that without explicit inclusion of the nuclear repulsion VN the total energy is a minimum at (or below) a = 90". When it is included, the energy minimum is close to a = 150". The corresponding value obtained from the values in fig. 3 is much nearer the experiment, being approximately 120". Similarly? a simple summation of calculated ionization potentials from fig.4 leads to a valence angle smaller than 90". * Since completing this manuscript we have noticed that Jsrgensen 8 has referred to this par- titioning in a recent book, but without any numerical applications. t In this connection it should be stated that we used both the coefficients and the integrals of Ellison and Shull, despite the fact that certain minor numerical mistakes in their work have been pointed out and corrected by McWeeny and Ohno.4 We have adopted this procedure because the errors are not large, and this facilitates comparison with Ellison and Shull's numerical work. The chief difference is that whereas the (21) curve in Walsh's diagram falls with increasing valence angle, the calculated (2al) curve rises.C. A.COULSON AND A . H. NIELSON 75 TABLE I.-SCHMIDTKE-PREUSS ENERGIES IN H20 (in eV) bond angle a 90" 1 20° 150" 1 80" total electronic energy 2 x 4 - 865.40 - 864.92 - 86443 - 864.14 2ZEjf VN - 614.34 -615.81 -616.22 -616.19 It is instructive to consider in more detail the essential difference between Walsh's energies in fig. 1, or the calculated ionization potentials in fig. 4, and our partitioned energies in fig. 3. As stated earlier this is the totally different behaviour of (3al). The core energy Ez. varies much more for this orbital-a change of over z a -Eat bond angle (a) FIG. 3. ? 52 - l o b - - 30 I bond angle (a) FIG. 4. 16.1 eV as the valence angle changes from 90 to 180O-than for any of the other orbitals. (The corresponding variations for the core energies of (1 al), (Zal), (I bz) and (lbl) are 0.1, 1.3, 10.0 and 0-0 eV respectively. The explanation of the greater variation for (3al) is the large contribution of the oxygen 2pa orbital; this has a large angular variation of the energy, unlike, for example, the Is that dominates (lal) or the 2s that dominates (2al).Table 2 shows these variations numerically.) TABLE 2.4ORE ENERGIES (-Ej) IN H20 (in ev) valence angIe a 90" 1 00" 105" 1100 120. 180" (1 ad 899.03 899.01 898.99 898.99 898.96 898-91 ( 2 4 214.12 21 3.73 213.72 213-56 213-73 212.86 (3~1) 188.45 192.19 193.32 194.08 196.75 204.60 (1bl) 204.60 for all angles W2) 167.57 165-48 164-77 163.85 161.84 157-5576 CORRELATION DIAGRAMS Now the value of c(J33-+K3J varies in the opposite direction by 9.79 eV. In the ionization potential, eqn.(3) shows that this sum occurs with a factor 2, but in the partitioned energy in (6) it occurs once. In (3) the variation of the Coulomb and exchange terms exceeds the variation of core energy ; in (6) it does not. This gives rise to the difference in slope of the two curves. Since the variation with angle of the Coulomb and exchange energy is important, it is interesting to show its precise value, and compare it with the sum of the ion- ization potentials (-2&), the sum of the core energies (- 2CEi) and the sum of the partitioned energies (- 2Zei = E- VN). Table 3 records these three quantities, and-in the third row-the value of 2C(&i-e$) = C(2Jij-Kij). In the range con- sidered this latter quantity varies more than any of the others except for the sum of the ionization potentials.It suggests that an absolute calculation of a valence angle by methods of this kind is by no means easy. This is, of course, precisely what has been found by Ellison and Shull and others. i i. j TABLE VA VARIOUS ENERGY SUMS IN H20 (inev) valence angle a goo looo 105" I loo 1200 180" - 2ZEi 1278.40 1275.52 1274.30 1274.54 127242 1262.66 - 2Zei 23 12-97 23 12.77 23 12-55 23 12.36 23 12.28 2309.84 2C(~i- ei) 1034.57 1037.25 1038-25 1037.82 1039.46 1047.18 - 2CEi 3347-53 3350.00 3350.79 3350.16 3351.74 3357.02 If it can be assumed that similar variations to those shown in fig. 3 will be found for all AH2 molecules, the importance of the nuclear repulsion term V' becomes apparent.For example, molecules of this type with four valence electrons are linear: yet both fig. 3 and fig. 4 would lead to an angle of approximately 90". It seems unwise, however, to pursue this argument since it would also predict a linear singlet CH2, and in any case the variation of the &i from the data of Krauss and Padgett 5 is entirely different from that shown in fig. 4 for H2O. It is probable that slightly different results from those shown in fig. 3 and 4 would result from improved molecular wavefunctions. But it is unlikely that our main conclusions would be materially affected. In order to test the generality of these results for H20 we are in process of ex- tending them to an AH3 system (e.g., H3O+), for which comparable s.c.f.6 and pseudo- Hamiltonian calculations 7 already exist. It is hoped to present the results of such calculations later. The chief drawback to the use of our partitioned energies ei is that they represent the influence of the orbital 4i in the presence of the other orbitals. If the number of these other orbitals changes, as by ionization or excitation, then the ei will also change. Perhaps, however, their general type of variation may not be much affected. We are now investigating this possibility. One of us (A. €3. N.) wishes to acknowledge the award of a Senior D.S.I.R. Fellowship, during the tenure of which part of the present work was performed. 1 Walsh, J. Chem. SOC., 1953, 2260, and following papers. 2 Schmidtke and Preuss, Z. Nuturforsch., 1961, 16a, 790. 3 Ellison and Shull, J. Chem. Physics, 1955, 23, 2348. 4 McWeeny and Ohno, Proc. Roy. SOC. A, 1960, 255, 367. 5 Krauss and Padgett, J. Chem. Physics, 1960, 32, 189. 6 Grahn, Arkiu Fysik, 1961, 19, 147. 7 Schmidtke, 2. Nuturforsch., 1962, 17u, 121. 8 Jsrgensen, Orbituls in Atoms and MoZecules (Academic Press, New York, 1962), p. 5.