J . Phys. Chem. 1990, 94, 5436-5439
5436 ACCURAFY
Figwe 1. Three-dimensional YPoplediagram” for present-day chemistry (adapted from ref 1). The non-self-avoiding nonrandom walk of John Pople through quantum chemistry space is shown on the accuracy sur-
face.
are required for the parts of the system involved in a reaction, for excited states and more generally for any phenomenon where changes in electronic structure play a significant role. However, even for such problems it is usually possible to treat only a small part of the system by semiempirical or a b initio quantum mechanical methods and to treat the much larger remainder, including the interactions with the quantum mechanical subsystem, by an empirical force field.’** The extended range of theoretical approaches just described makes it useful to add a third dimension to the Pople diagram (see Figure 1). In addition to sophistication (type of method) and complexity (number of electrons), there is needed a dimension (7) Singh, U. C.; Kollman, P.J . Compur. Chem. 1984, 5, 129. (8) Bash, P.A.; Field, M.J.; Karplus, M.J . Am. Chem. Soc. 1987, 109, 8092-8094.
that provides an estimate of the accuracy of the calculation for the system under consideration. A simple proportionality between sophistication and accuracy, implied by the two-dimensional Pople diagram, is not applicable to the wide variety of calculational methods now being employed. An impressionistic view of the accuracy of the various methods is given by the vertical dimension in Figure 1. This yields a limiting surface for the accuracy of a given type of calculation for a given system size. The edge of this surface projects onto the hyperbola of quantum chemistry, with the two axes in the plane corresponding to those of the Pople diagram. On the sophistication axis, only empirical methods have been added and they have been assumed to be least sophisticated (in deference to the quantum chemistry audience at this meeting). As to the size of systems to be studied, the linear scale in the Pople diagram, which covered the range from 1 to 100, has been replaced by a logarithmic scale that goes from 1 to lo6. The accuracy of a method increases with distance from the Pople plane. All methods, at whatever level of sophistication, are presumed to reduce to the exact result for one electron. Beyond that, the best a b initio and the empirical methods give the highest accuracy, with the former extending to systems of 10’ electrons and the latter to l@ electrons or more. The other methods fall in between these limits of sophistication and number of electrons with the accuracy surface being rather complex. For the a b initio methods, there is a monotonic increase in sophistication and accuracy and a monotonic decrease in range of applicability in going from minimum basis LCAO-SCF, to extended basis (full) Hartree-Fock, to low-level and high-level correlation calculations. The evaluation of the accuracy dimension in the semiempirical direction from the LCAO-SCF methods is more difficult. The drawing suggests that PPP/CNDO-type methods, which generally make use of a minimal basis set, are more accurate than minimal basis set LCAO-SCF ab initio calculations and also are more accurate than Hiickel calculations. Not included in the diagram are density functional methods, which appear to violate the hyperbola of quantum chemistry. They are in the range of accuracy and sophistication of Hartree-Fock-type calculations but can treat a larger number of electrons with the available computer time. Since this meeting and issue are dedicated to John Pople, I have tried to encompass his work over four decades by drawing a path representing his work in the quantum chemistry space in Figure 1. H is wide-ranging interests and research accomplishments make clear that, even today, a single person can span groups I and 11.
John Pople: The CNM) and INDO Methods Gerald %gal College of Letters, Arts and Sciences, Administration 200, University of Southern California, Los Angeles, California 90089-401 2 (Received: November 10, 1989)
A reminiscence on the development of the CNDO and INDO methods.
In 1963, John Pople chow to leave England and join the faculty of Carnegie Mellon University, then Carnegie Institute of Technology. After some visa problems, he actually arrived in March of 1964 and I was there, a first-year graduate student, expectantly lying in wait for him. Thus the honor fell to me to become John’s first American graduate student. David Santry had come with him from England and Mark Gordon joined the group a few days later; my honor was short-lived. Most of you know John today after twenty-five years of acculturation to this country, so you can well imagine that our lab routine was more than a little English. Each day the four of us had afternoon tea. 0022-3654/90/2094-5436$02.50/0
In those hours of relaxed conversation, I learned enormous amounts of quantum chemistry from John, and I have always felt lucky, lucky to have been with him at a time when he was new to the country and thus not too busy and lucky to have worked with one of those rare men who are so good with students that their products form an entire school of science. John Pople’s students are everywhere today and each of them, I think, feels as fortunate as I to bear the marks of his influence. When we sat down to discuss what I might do, John suggested that it would be good to try to extend the neglect of differential overlap approximation, the Pariser-Parr-Pople theory of calcu0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5437
John Pople: The CNDO and INDO Methods lations on the ?r electrons of planar conjugated molecules, to general geometries, i.e., to u-bonded systems as well. The essence of the approximation in PPP theory is that the Hartree-Fock equations are taken to refer to an orthonormal basis set so that the overlap matrix is the identity matrix and the equations to be solved are FC = CE (1) where F is the Fock matrix, C the matrix of molecular orbital coefficients, and E the energy. The expression for the matrix elements of F is
where H,,, are the one electron matrix elements, P is the charge density and bond order matrix, and (pvlAu) are the two electron re ulsion integrals. If the molecular orbitals are normalized to = 1, then the sum of the charge densities, the diagonal elements of P,add up to the number of electrons in the molecule and the classical number of electron repulsions will be correctly counted in the diagonal matrix elements if the two electron repulsion integrals are taken as if p # v or A # u (3) (pvlAu) = 0 while a remanent of the exchange energy exists in both the diagonal and the off diagonal elements of the Fock matrix. It seems good to me to remind you of the time. Roald Hoffman's paper on extended Hiickel theory' appeared in 1963. Gaussian methods were just getting started; POLYATOM was in existence and a calculation on the ethylene ground state using a Gaussian basis set2appeared at about the same time as the CNDO method3 which was first presented at the Sanibel Island meeting in honor of Robert Mulliken in January of 1965. The computer at Carnegie Tech was a Bendix G21, the language of choice there was ALGOL, and configuration interaction methods were in their infancy while no practical MCSCF approach existed. When we began to look at extending the neglect of differential overlap approximation to the general case, it quickly became apparent that there is a strong difference from the situation in a systems where, by definition, the orbitals are all parallel. Here, given an inertial coordinate system, the placement of the molecule implies some mixing of the original orbitals; that is, given a p,, orbital on an atom A pointing directly along some coordinate toward an atom B, rotation of B by 45" about A produces
C'k
two-center integrals as in CNDO. This is the intermediate neglect of differential overlap approximation (INDO). 3. In addition to those integrals retained in INDO, maintain integrals which involve off-diagonal distributions in which two orbitals on one center interact with a distribution in which the other orbitals are also on a common, but different, center. This was called the neglect of diatomic differential overlap approximation (NDDO). At this point, John chose to develop the simplest possible approximation, CNDO. The objective was to find a semiempirical approximation which was completely defined, Le., which did not allow for adjustable parameters, once a canonical parametrization had been found. It was, of course, possible to choose the canonical parametrization to provide an optimal fit to some set of experimental observations or to choose it so as to provide a fit to the underlying theory, ab initio Hartree-Fock theory. The latter was chosen. I think that most of you will recognize this philosophical approach in John's subsequent work with Gaussian basis sets which did so much to systematize our field. A parametrization was worked out and the results seemed to be extremely promising in that not only could reasonable agreement be found with ab initio wave functions and orbital energies for polyatomic molecules like NH3, BH3, CH4, and HCN, but the theory also provided reasonable molecular electric dipole moment predictions and a moderately satisfactory barrier to internal rotation in ethane. This was the theory presented in January of 1965 at the Sanibel Island Symposium in honor of Mulliken, a now defunct version known as CNDO/l; however, time had been short, the power of the theory had only been partially tested, and we believed that the parametrization could be improved upon. We returned to Pittsburgh and went to work on an improved version, CND0/2, which differed from the previous theory in only two regards. 1. In the previous theory, the appropriate collection of integrals within the diagonal Fock matrix element had been approximated by the appropriate experimental atomic orbital ionization potential. Now the average of the experimental ionization potential and electron affinity was used under the philosophy that the theory sought to represent atoms in both positive and negatively charged situations and an appropriate average should be struck. 2. In the earlier theory the integral representing the attraction of a nucleus A for an electron in an orbital on another center B, VAB,had been a computed quantity; it was now approximated as
(4)
and the density distribution becomes Now the essence of the neglect of differential overlap approximation is that off-diagonal charge distributions are neglected, so that if one considers Coulombic interaction with some arbitrary charge distribution on atom B, pB, (Pup,lpB)is inequivalent to (p,'p,'JpB) because of the neglect of the off-diagonal distribution p g I . The calculation is not geometrically invariant so that, for instance, one cannot calculate molecular geometries without inherent error as one moves the atoms around. Correcting of this implied a number of possible levels of approximation. 1. Make all Coulombic interactions a function of the atoms alone and not of the nature of the interacting orbital interactions. There are then only a set of Coulombic interaction parameters characterized by their atoms, r A B . Acronyms were already endemic in quantum chemistry and this was called complete neglect of differential overlap, CNDO. 2. Maintain integrals which involve off-diagonal distributions on one atom interacting with others on that center, but keep all (1) Hoffmann, R. J . Chem. Phys. 1963.39, 1397. (2) Moskowitz, J.; Harrison, J . Chem. Phys. 1965, 42, 1726. (3) Pople, J. A.; Santry, D. P.;Segal, G. A. J . Chem. Phys. 1%5,43, S129. Pople, J. A., Segal, G. A. [bid. 1965, S136. (4) Poplc, J. A. J . Chcm. Phys. 1965, 43, S229.
where ZBis the atomic number less 2, approximating complete shielding by the 1s electrons. The result is an approximate expression to the matrix elements of the Hartree-Fock which is pleasing for the simple physical elegance of the expression of the diagonal FA, = 4.5(1,
+ Ah) + [(Pm- Z A ) - 0.5(pAA - i)]r, +
E (PBB- Z B ) ~ A(7) B
BZA
The diagonal matrix element is the Mulliken electronegativity (fitting since CNDO was first presented at a meeting honoring Robert Mulliken) which is a fundamental expression of the energy binding an electron in orbital A on atom A, corrected by electrostatic terms. (Pm - Z,) is the net charge on the atom and vanishes if the atom is neutral. (PAX- 1) vanishes if the orbital is occupied by only one electron and the last term vanishes if all the other atoms are electrically neutral. Now a feature of John's work which was to become extremely obvious in his Gaussian work came into play. He told me to test the theory by calculating the predicted geometry, dipole moment, and bending force constants for all possible AB2 and AB3 molecules involving only the atoms of the first row and hydrogen, throwing in for good measure a few electronic excited states for which variational principles exist. Today, I could readily do this in a day on a workstation on my desk, but then it was quite a lengthy task and I could only get enough computer turnaround to get two
Segal
5438 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 TABLE 1: Summary of Rcrults for A& Mokeules no. of valence electrons 4 5 6 7
8 9 15 16 17
18 19 20
B-A-B angle, deg molecule
calc 180.0 136.6 180.0 108.6 141.4 107.3 145.1 118.7 107.1 180.0 180.0 180.0 180.0 180.0 180.0 142.3 137.7 124.6 118.3 114.0 104.6 102.5 99.2
obs
dipole moment, D calP
obsM
0
0.51 0
103.2b 1806 103.3c 144d 1O4.4Sc 18V 1 80h 186 180' 180' 134m 132" 115.4' 116.8 (100 or 108)" 104.2" 103.8w
2.26 0.75 2.16 0.87 2.08
1.8Y
0 0 0 0
-0.75 0.05
f0.4O
-1.26 0.53 -0.12 -0.21
f0.58' f0.297"
bending force const, mdyn/A calc 0.07 0.27 0.23 0.69 0.38 0.81 0.34 0.52 0.951 0.19 0.36 0.44 0.58 0.12 0.44 0.73 0.66 0.62 1.13 0.76 I .09 0.97 0.71
obs
0.69' 0.26* 0.57f 0.7Y 0.42' 0.4W (0.58 ) q 1.755 I .2sr (1.28)q 0.59
P00
1.144 1.040 1.015 0.991 0.937 0.9 17 0.849 0.591 0.856 0.968 6.134 5.823 6.266 7.271 5.883 6.528 6.204 7.162 6.543 6.152 7.145 7.113 7.060
r/, A 1.343' 1.180' 1.18W 1.094' 1.094' 1.024' 1.024' 0.96W 0.96W 0.92W I .25W 1.176' 1.162 1.36W 1.154 1.200' I .200 1.300' 1.236 1.278 1.320' 1.350 1.410
Tables of Interatomic Distances and Configuration in Molecules and Ions; The Chemical Society: London, 1965. Herzberg, G . Proc. R . SOC. (London) 1961, A262,291. 'Drtssler, K.; Ramsay, D. A. Philos. Trans. 1959, A251, 553. dDixon, R. N. Mol. Phys. 1965,9,357. 'Posener, D. W.; Strandberg, M. W. P. Phys. Reo. 1954, 95, 374. fHerzberg, G. Infrared and Roman Spectra; Van Nostrand: New York, 1945. ESommer, A,; White, D.; Linevsky, M. J.; Mann, D. E. J. Chem. Phys. 1963, 38, 87. hRamsay, D. A. Adu. Spectrosc. 1959, 1 . 'Brewer, L.; Somayajulu, G . R.; Bracket, E. Chem. Reo. 1963, 63, 1 1 1 . 'Buchler, A.; Klemperer, W. J. Chem. Phys. 1958, 29, 121. 'Steeman, J. W. M.; Macgillavry, C. H. Acta Crysrallogr. 1954, 7, 402. 'Teranishi, R.; Decius, J. C. J. Chem. Phys. 1954, 22, 896. mOvenall, D. W.; Whiffen, D. H. Mol. Phys. 1961, 4, 135. "Claesson, S.; Donohue, J.; Schomaker, V. J . Chem. Phys. 1948, 16, 207. OZahn, C. T. Phys. 2.1932, 33, 686. PHisatsune, I. C.; Devlin, J. P.; Califano. S.Spectrochim. Acta 1960, 16, 430. qNagarnjan, G . Aust. J . Chem. 1963, 16, 717. 'Carpenter, G . B. Acta Crystallogr. 1955, 8, 852. 'Weston, Jr., R. E.; Brodasky, T. F. J . Chem. Phys. 1957, 27, 683. (Hughes, R. H. J . Chem. Phys. 1953, 21, 959. "Milligan, D. E.; Mann, D. G.; Jacox, M. E. J . Chem. Phys. 1964.41, 1199. "armony, M. D.; Myers, R. J. J . Chem. Phys. 1961,35, 1129. "Ibers, J. A.; Schomaker, V. J. Phys. Chem. 1953, 57, 699. xPierre, L.; Jackson, R.; DiClanni, N. J. Chem. Phys. 1961, 35, 2240. YMcClellan, A. L. Tables of Experimental Dipole Moments; W. H. Freeman: San Francisco, 1963. 'Assumed average bond length. 4aDipole moment + in the direction A-B+.
or three points a day. The results were very exciting as they developed and John would come by my desk each morning to get the latest results. There was a blackboard next to the desk and soon I made up into a kind of scorecard which became Table I, which we published in the original5 paper. The angles were very exciting because they fit experiment closely with but two glaring exceptions: that for the first excited state of NH2, a state whose optical spectrum exhibits a Renner-Teller splitting. N H 2 was predicted to have an equilibrium bond angle of 145.1 but was reported to have a 180-deg equilibrium angle. The potential energy curves calculated by CNDO for the Renner-Teller states is reproduced in Figure 1. Then, just as I was finishing up the table, a piece of work by Richard Dixon appeared6 in which he reanalyzed the spectrum and found the equilibrium angle to be 1 4 4 O . To my mind, this was the first true prediction of an experimental quantity by CNDO. The first triplet excited state of C H 2 was also predicted to be bent while experiment seemed to indicate that it was linear. History, after a great deal of work and debate by many people,' has borne out the CNDO prediction and the bond angle of this state is thought to be about 134'. This is the CNDO theory which lives today. It is, however, physically defective. Consider the off-diagonal matrix element. The second term on the left is a remanent of the exchange energy in the CNDO approximation. In a planar molecule, if p is u and Y is r , then the bond order and @ are identically zero by symmetry. No exchange integral appears on the off diagonal. It turns out that this leads to an inability in the theory to predict spin po( 5 ) Pople, J. A.; Segal, G. A. J. Chem. Phys. 1966,44, 3289. ( 6 ) Dixon, R. N. Mol. Phys. 1965, 9, 357. (7) Schaefer 111, H. F. Science 1986, 231, 1100.
- - 12.860 - i2.920 c
30
/
f00
\
120 140 I60 (80 200 220 240 260 280 9
Figure 1. Energy on bending NH2.
larization in the u system from a r radical as well as a number of other inadequacies in physical prediction, notably with respect to magnetic properties. The solution is just the integrals left out of CNDO which are included within INDO, so there are strong physical motivations for movement to the next level of approximation. The INDO method was immediately and relatively simultaneously developed by Pople, Beveridge, and Dobosh8 and by D i ~ o n . ~It is a fuller theory which retains the successes of CNDO while allowing for a number of new effects. While the utilization of CNDO and INDO quickly became widespread, there remained, of course, the alternative philosophical approach in which the fundamental theory is parametrized not to approximate ab initio theory, but to some set of experimental parameters, and this approach was also quickly taken up, although (8) Pople, J. A.; Beveridge, D. L.; Dobosh, P. A. J. Chem. Phys. 1%7,47, 2026. ( 9 ) Dixon, R . N. Mol. Phys. 1967, 12, 8 3 .
J . Phys. Chem. 1990,94, 5439-5444 not by the Pople group which adhered to the philosophy of constructing a well-defined model which was then to be understood for its strengths and limitations. Thus Del Bene and Jaffe quickly moved to a parametrization CNDO/S'O which was particularly well suited to the calculation of electronic excitation energies and oscillator strengths for conjugated systems including the effects of polarity in heteroatomic molecules. Similarly, Dewar" and =workers immediately put forth the first of what were to be many generations of M I N D 0 methods which sought develop a theory based upon INDO which would allow one to reliably to predict heats of formation. This, of course, has betn extremely successful and is widely used today. It is an available option in computer programs like GAUSSIAN 86. John, however, moved on to a b initio methods, and his group produced GAUSSIAN 70, which was to quantum chemistry what the IBM PC was to personal computers-it organized and formed (IO) Del Bene, J.; JafE, H. H. J . Chem. Phys. 1968.48, 1807; 1968,49, 1221. (11) Baird, N. C.; Dewar, M. J.
S.J . Chem. Phys. 1968,50, 1262.
5439
the field. That transition, however, is the subject of the next talk and it is time for me to finish. John Pople has been a leading figure in theoretical chemistry from the first classic papers published by the Lennard-Jones group at Cambridge in the early 1950s, through the N M R years, to his seminal effect on computational quantum chemistry. During the time that I was with him, John received an award-I cannot remember what it was and John cannot remember either. At any rate, when I congratulated him on it, it was obvious that the fact that he was just turning forty was bothering him, because he answered, "When this sort of thing starts happening, you know you're getting old". (Let me tell you, when somebody asks you to give an historical talk on your thesis, you know you're getting old.) In John's case, we know that he had nothing to fear-he was to be a fountain of new and imaginative science for the next twenty-five years and today we honor the forty years of his career so far. As I said in the beginning, I feel fortunate to have been one of those to benefit directly from his influence and I want to thank Fritz and Nick for giving me the opportunity to dwell on an important piece of work in John's career and an important period of my life.
John A. Pople: Early ab Initio Days Leo Radom Research School of Chemistry, Australian National University, Canberra, A.C.T.2601, Australia (Received: November 9, 1989)
An account is presented of research activities in John Pople's group at Carnegie-Mellon University from 1968 through 1972. The development of GAUSSIAN 70 during this period meant that ab initio calculations could be carried out straightforwardly and rapidly. This opened the way for applications of ab initio theory on a scale that would previously not have been feasible and laid the foundations for the widespread use of ab initio calculations by the chemistry community.
When I received the invitation to speak at the symposium Forty Years of Quantum Chemistry: An International Conference in Honor of Professor J. A. Pople, I had mixed emotions. On the negative side, this was the first occasion on which I had been asked to give a historical talk, and it made me realize that time does not stand still and that age must be catching up with me. On the positive side, I was delighted to be able to contribute to a symposium that deservedly honors my mentor, colleague, and friend, John Pople. So thank you to the organizers, Fritz Schaefer and Nicholas Handy, for having provided me with the opportunity to participate, and congratulations to them also for having brought together a t this conference such an impressive array of talent. My brief in this presentation is to discuss "the early a b initio days". I will be examining a period of roughly 5 years from 1968 through 1972. I should stress at the outset, so as not to offend anyone, that I am not attempting to review the advances in quantum chemistry that took place outside of Pittsburgh. This would be an impossible task in the time available. Rather, I will be focusing entirely on events in the Pople group a t CarnegieMellon University during this period. I spent 3l/* of those years beginning in early 1969 as a postdoc in the group, and a very pleasant and stimulating time it was. But the story begins before I arrived in Pittsburgh, and we should probably take it up in early 1968. At that time, the series of all-electron semiempirical S C F schemes were already firmly established' and were offering an economical alternative to the a b initio calculations that were being carried out in various laboratories, mainly in the U.S.but also elsewhere in the world. A (1) For a review of the semiempirical procedures. see: Pople, J. A.; Beveridgc. D. L. Approximate Molecular Orbital Theory; McGraw Hill: New
York, 1970.
ballpark figure of about 1:lOOO would have described the relative computing costs of say CNDO and minimal basis set a b initio calculations at the time. John Pople saw a natural progression from the semiempirical to the ab initio approach. As a first step in attempting to bridge the gap, he introduced, together with Marshall Newton and Neil Ostlund, the approximate ab initio PDDO (or projection of diatomic-differential-overlap) schemee2 This involved approximating some of the more problematical integrals using a generalization of the Mulliken approximation. The PDDO method was slower than CNDO (by a factor of about 10)but produced useful savings in computation time compared with the fully ab initio approaches that were available at the time. However, important developments in the fully ab initio procedures soon took up the running. They had their origins in the work of Boys, who, some 20 years earlier,3 had suggested that the integral bottleneck in a b initio calculations, the evaluation of four-center two-electron integrals of the type (4&1J#J~&)involving basis functions &, could be overcome by using Gaussian basis functions (exp(-d)) instead of exponential basis functions (or Slater-type atomic orbitals, STOs). The Gaussian functions had the crucial property that the product of Gaussian functions at two centers A and B was itself a Gaussian function at a third center C. The disadvantage of using Gaussian basis sets was that several Gaussian functions were needed to reproduce the results of a single (2) (a) Newton, M. D.; Ostlund, N. S.;Pople, J. A. J . Chem. Phys. 1968, 49, 5192. (b) Newton, M. D. J . Chem. Phys. 1969, 51, 3917. (c) Newton, M. D.; Lathan, W. A.; Hehre, W. J.; Pople, J. A. J . Chem. Phys. 1969, 51, 3921.
(3) Boys, S. F. Proc. R.Soc. London 1950, A200, 542.
0022-3654/90/2094-5439$02.50/00 1990 American Chemical Society
