Written By Rod Bartlett
When most read a popular account of scientific progress, the focus is on the ‘big name’ projects that one knows from the press: solar energy storage, hydrogen fuel, reduction of greenhouse gases, cures for cancer, etc. And the ‘experimental’ tools used to address these issues that measure the success or failure of some hypothesis. That, after all, is the scientific method. But as observations are made, science is trying to construct an underlying, organizing ‘theory’ that explains the experiment and will explain untold other future observations. An example is the difference between Newton showing that a prism splits light into many different colors—an experiment—and deriving the equations from his very general laws that explain the observed optics of the prism. The latter enable ‘predicting’ untold other optical phenomena in the absence of experiment.
Therefore, in this case Sherlock’s admonition that it is ‘dangerous to theorize without the facts’ needs some modification for ‘predictive’ theory. When the equations are correct and can be solved, the results have to be true. Today, that kind of predictive theory is what has been developed by quantum mechanics that in Dirac’s phrase underlies ‘all of chemistry.’ Except in his opinion, ‘the equations are too difficult to be soluble.’ The latter is no longer true. All those highly visible ‘big name’ projects depend upon chemistry, and chemistry deals with in Mulliken’s phrase, ‘what the electrons are really doing in molecules.’ With this knowledge, the energies of reactions, the activation barriers that control what reactions occur, and the spectroscopic fingerprints that identify the molecules become known.
The description of electrons requires the solution to the familiar, quantized equations of quantum mechanics for the electronic ‘wavefunctions’ and their energies, HΨk=EkΨk. But the H in these equations describe the Coulombic interactions among a molecule’s ‘many-electrons’. That means the water molecule’s 10 electrons produce a ‘10 body’ problem (45 electron-electron interactions), or for benzene, a’46 body one’ (1081 interactions), or for a piece of DNA, many more. Yet, we can only solve the Schrodinger equation of QM exactly for 1 electron, the hydrogen atom. So, we are faced with having to develop mathematical and computational tools that allow sufficiently accurate solutions of such many-electron problems to obtain the secrets of the molecules in question. When we are able to do that, we have a direct route to facts that are not typically amenable to experimental observation, like for molecules under extreme conditions as in explosions, or in interstellar space, or the detection and identification of rocket plumes, or the design of new concepts for fuels, among many other applications. Providing these solutions is the science of quantum chemistry.
But one major problem remained in its application. The problem of ‘electron correlation’. Electrons are charged particles meaning they interact instantaneously through Coulomb forces that cause their motions to be ‘correlated’, and these interactions are missing from an average (‘mean-field’ approximation) like the well-known Hartree-Fock theory. The latter approximates Ψ0 by Φ0, the familiar molecular orbital approximation that provides the conceptual interpretation of much of chemistry. Quantum chemical solutions to define Φ0 have been practical for many applications since the sixties, but the relatively small ‘correlation’ contribution that distinguishes the correct solution is critical to a ‘predictive’ theory for bond energies, activation barriers, spectra, and structure, indeed chemistry. As such, it has been the dominant unsolved problem in quantum chemistry for about 50 years.
In our forty years of AFOSR support, a number of notable advances have been made in the solution of the correlation problem. As a young scientist at Battelle in Columbus, Ohio, I approached Ralph Kelley, an AFOSR program manager in physics about support. I told him about using many-body perturbation theory (MBPT) and its diagrammatic framework borrowed from quantum field theory and Feynman diagrams to treat ‘electron correlation.’ He and AFOSR enabled me to start as an AFOSR PI in 1978.
As a postdoc at John Hopkins with Robert Parr, I had been given the freedom to pursue the many-body theory I had begun as an NSF postdoc at Aarhus, University in Denmark, in 1973. I and my collaborator, David Silver from the Hopkins Applied Physics Lab had written the first papers in chemistry in 1974-76 showing the potential power of MBPT. Prior work was due to Hugh Kelly in physics, who applied MBPT to atoms, but molecules require a very different treatment, so these were the first such applications. The reason it is called many-body perturbation theory (MBPT) is that the theory is based on the linked-diagram theorem of Brueckner and Goldstone that guarantees correct scaling with the number of electrons. Linked diagrams describe the electron-electron interactions in its most compact way. The energy of one of these quantum states has to be ‘extensive’ , so it should grow correctly with the number of electrons, a feature we later termed, ‘size-extensivity’ as the rationale for all many-body treatments. Although it should be obvious that when all the units (or atoms) in a molecule are too far apart to interact, the correct energy should be the sum of the energies of the units; but this condition is not met by the variational, configuration interaction approximations that were in dominate use during those 50 years. The many manifestations of size-extensivity were not to be fully realized until the turn of the century. Today, it is deemed a fundamental property that all worthy electronic structure approximations should satisfy.
Two years after our initial MBPT papers, John Pople decided to apply this method, but chose to call it Moeller-Plesset perturbation theory (MPPT) as he tried to avoid the less familiar diagrammatic tools we used. But his terminology hides the fundamental rationale for these many-body methods, in that the identification of ‘linked diagrams’ guarantees size-extensivity, and this feature is not apparent in ordinary perturbation theory. Today, the MBPT=MPPT methods for solving the Schrödinger equation are in virtually all quantum chemical programs. A search of the Web of Science shows that though there were only a hand-full of citations in the 70’s, and a couple of hundred until ~1989, there are now more than 295,700 citations to the method and 8105 papers written about it.
But perturbation methods are limited to some order and since the correlation correction is not small (in extreme forms it accounts for phase transitions in solids and super-conductivity), a far more powerful many-body approach is to sum many such linked diagram terms that describe correlation to infinite order. This is the idea of coupled-cluster theory that shows that the correct, infinite-order MBPT wavefunction for any system, Ψ0 = exp(T)|Φ0>. The exponential form guarantees size-extensivity. The cluster separation of, T=T1 +T2 +T3 +…where the subscripts indicate one-electron, two-electrons, three-electron,… provides a framework for a wealth of approximations determined by the number of clusters retained, like CCSD for single and double ones. The size-extensive property is at work at any truncation of T, providing superior solutions to any that had been previously obtained for the same computational effort. This is because the CC wavefunction even limited to T2, the double excitation cluster operator, automatically has all products like ⅟2T22 which are ‘quadruple’ excitations, and ⅟6 T23, ‘hextuple,’ etc., in its wavefunction. As T1, and T3 are added, one rapidly exhausts the effects of electron correlation converging to the exact solution.
The first report of general applications of CC theory, i.e. CCD for just T2 were reported in 1978 by us and Pople in back-to-back papers. Then George Purvis and I first reported CCSD (CC for single and double excitations) in 1982. Our CC papers were supported by AFOSR.
Because of the products included in exp(T2), unlike CI, most ‘quadruple’ effects are already included in CCSD, so the next most important term in due to T3. In our next AFOSR work (1984) we reported the first general inclusion of triple excitations (CCSDT-1), followed by the first non-iterative approximation CCSD[T], in 1985. A better non-iterative approximation, CCSD(T), that added one small term to [T] was introduced by the Pople group (1989) without a rigorous derivation. We presented that in 1993. The latter is now called the ‘gold standard’ in quantum chemical calculations. In 1987 we reported the full CCSDT method for the first time, sometimes called the ‘platinum standard’, followed later by full quadruples, CCSDTQ, and pentuples, CSDTQP! In this way we were able to show the rapid convergence of CC theory to the exact result, documenting its predictive character. Another citation check shows that from virtually no mention in the seventies, to less than a hundred citations in the eighties, CC theory has now spawned 28,780 papers and over 700,000 citations.
Another advantage that calculations have over experiment is the flexibility of application. In a second project with AFOSR, the physics program manager with responsibility for non-linear optics (NLO), Col. Gordon Wepfer, showed me experimental results for electric-field induced second and third harmonic generation experiments in the gas phase, compared to the theory of the time. The theory was hopeless! NLO effects are critical to all kinds of problems from protecting pilots’ eyes from lasers to doing selective surface chemistry. They are in principle, amendable to quantum chemistry, as they depend upon the higher terms in the expansion of a molecule’s energy in the presence of (frequency dependent) electric fields. These quantities are called hyperpolarizabilities, as they are higher-order generalizations of the well-known dipole polarizability for a molecule. I could not promise that we could resolve the discrepancy between theory and experiment, but with our new CC/MBPT methods I could promise to do calculations for such quantities with the best correlated quantum chemistry that existed. It took a few years and required some new theory for the treatment of the frequency dependent effects, but, indeed, we were able to explain the observed experimental values for the first time.
Another illustration of the flexibility of application occurred when Capt. Pat Saatzer of the Rocket Lab asked us at Battelle to provide a theory complement to two experimental efforts, one directed by John Fenn, later to be a Nobel Laureate, to determine the cross sections for vibrational excitations when components of combusted fuel collide with O atoms in the upper atmosphere. The idea is that depending upon the products in the fuel, a knowledge of these signatures allows one to identify whose missile it is. This kind of problem requires the combined efforts of molecular dynamics and quantum chemistry, the latter to provide the potential energy surface of interactions between the molecules and O atoms, and the dynamics, done by Mike Redmon, to add the time-dependent aspects. Both experiments failed, leaving only the theory to provide the cross sections required in the deployment of detectors.
A third illustration deals with NMR spectra. NMR has two components, a vector term that gives the chemical shift and a scalar term that provides the J-J spin-coupling constants for molecules. As the latter connects any two atoms in a molecule through its electronic density, the analogy with a chemical bond has inspired a lot of discussion. Once again, inspired by the lack of agreement between theory and experiment as pointed out in some review articles, Ajith Perera and I decided to apply our new CC/MBPT tools to resolving this issue. Once again these methods were remarkably successful, providing the first ‘predictive’ theory of J-J coupling constants. We went on to use them to further resolve the long-term argument between Georg Olah and H. C. Brown about the existence of non-classical C bonding, and with Janet Del Bene, to study the ‘two-bond’ coupling across an H-bond in nucleic acid bases. By measuring the latter, one can infer the location of the H- bond that cannot be seen in X-ray analysis. We also offered a J-J signature for the meaning of a strong, weak, or normal H-bond.
The next major theory effort for AFOSR was our further development of CC/MBPT but now focused on excited states. In the Schrodinger equation above, the ‘k’ indicates one of the many quantized solutions to the problem. The others are important to electronic spectroscopy and photochemistry among many other needs. In its original formulation, CC/MBPT provided very accurate results for one state, but we changed that by introducing what we call the equation-of-motion (EOM) CC starting in 1984-1992. This enables one to add a spectrum of excited states on top of a CC solution for the ground state. EOM also permits ‘excited’ states that differ from the ground state in the number of electrons, as in ionizations in photoelectron spectroscopy (IP-EOM-CC), or by adding an electron (EA-EOM-CC), or kicking out two electrons, DIP-EOM-CC or adding two, DEA-EOM-CC. Hence, one now has a wide array of ways to describe ‘what the electrons are doing in molecules’ for a wealth of different situations. Subjecting EOM-CC (sometimes called CC linear response) to the same measure of use as the other two developments shows over 23,500 citations and 690 papers using these methods today.
Armed with all these tools, a fascinating problem arose in the new, high-energy density material (HEDM) program geared toward new ideas for ‘revolutionary’ improvements in rocket fuels. I submitted a proposal entitled “An Investigation of Metastability in Molecules” to Drs. Larry Curtiss and Larry Burggraf in AFOSR chemistry, that asked the question how much energy could be stored in a molecule with a sufficient barrier to decomposition to keep it around long enough to be useful. Later Dr. Mike Berman became the program manager, and remains my program manager today. My proposal planned to use our predictive set of quantum chemical tools to address this question. Unlike synthesis, which is difficult, expensive, and dangerous, quantum chemical applications can explore prospects that exhibit different principles to see if any might be worthy of further study.
One strategy for storing energy into molecules would be to force some atoms to bind in unfamiliar ways, a concept we termed ‘geometric metastability.’ A case in point is the tetrahedral form of N4. As the normal form of P4 is a tetrahedron, and N and P are isovalent such a molecule makes sense. But while P2 is not very stable compared to P4, the N2 triple bond is one of the strongest bonds known, and four N atoms energetically prefer two N2 molecules to four single bonded N atoms in a tetrahedron. That, of course, is exactly what one would like, since if the four N atoms could be put into a tetrahedron, and if there is a barrier to decomposition that would keep it around; then under stimulus all the energy in N4 could be released to N2 molecules. Our calculations show that N2 would release 190 kcal/mol and would be held together by a barrier of 40 kcal/mol, once the four atoms could be put into the tetrahedron. That, of course, is the difficulty. Although there have been some potential experimental observations, perhaps the best one is from mass spec, where its isoelectronic analogue, N3O+ has been seen.
Another of our predictions was the existence of the N5– pentazole anion. Again, this makes perfect sense in terms of its bonding, even achieving extra stability via its pi electron aromaticity like in benzene. In this case we predict a barrier to decomposition of 27 kcal/mol. It has now been observed in negative ion mass spec as a byproduct of a known pentazole containing molecule. The targets for the HEDM project originated with theory that spun-off further work by DARPA and NASA, with the former pursuing serious synthetic efforts. Recently, another of our predictions, N8, seems to have been seen experimentally. Some have also been seen in high pressure experiments
Everyone in the computational field would love to be able to make accurate calculations by using an effective one-particle theory, so that all the complicated two-particle terms that must be described in CC theory could be avoided. This is the impetus for the development of Kohn-Sham density functional theory (KS-DFT). But unlike CC, there is no way to converge to the right answer, since the correct density functional is not known in any useful way. Instead, thousands of density based approximations to the KS-DFT theory are made and used to get answers quickly, without any guarantee of veracity.
In our current work for AFOSR we have tried to improve upon such an approach by insisting upon a rigorous foundation. That foundation starts by our formulation of a ‘correlated orbital theory’ (COT). It was derived from manipulating the IP/EA-EOM-CC equations that are formally exact into an effective one- particle form, whose eigenvalues have to correspond to the energy required to remove any electron from the molecule (IP) or to add an electron to the molecule (EA). This approach augments the mean-field Hartree-Fock approximation with a correlation orbital potential (COP)!
Since KS-DFT is a special case of COT, using this rigorous theory as a model, one can assess the accuracy of various DFT approximations. Finding that none satisfy our conditions, we took some well-known forms and, by virtue of the 2-4 parameters in them, fit them to satisfy our eigenvalue property initially only for water’s five Ip’s. In this way we introduced QTP(00). Two new minimally parameterized approximations, QTP(01), and QTP(02) followed. All provide accurate one-particle spectra to some threshold from the eigenvalue attached to each MO, proven by testing them against 401 experimental values from 63 molecules. QTP(01) gets all valence IP’s accurate to ~10%. An important application is core ionization and excitation, where QTP(00) is without peer. It accurately describes the core spectra of all the amino acids. Unlike any other DFT approximation, QTP(02) correctly describes the EA, both bound and unbound. The QTP family also gives excellent activation barriers, excited state excitation energies, and the molecular densities, themselves. As the avowed goal of DFT is to provide accurate densities, the QTP functionals do that better than most.
This QTP family defines what we call ‘consistent’ KS-DFT approximations, since one cannot get the IP-eigenvalues right without a good KS potential. Also, the connection between the orbital eigenvalues and Ip’s requires that the excitations given by adiabatic time-dependent DFT (TDDFT), be correct for excitation into the continuum, i.e. ionization. Further, an accurate potential mitigates the debilitating self-interaction error of KS-DFT, where electrons incorrectly interact with themselves. When we insist upon ‘consistency,’ we are a step closer to our goal of mimicking the predictive results of CC theory in a highly efficient one-particle theory. This is another testament to the CC revolution that began and was nurtured by AFOSR!
Besides the AFOSR work mentioned here, it is important to recognize that other aspects of our formative many-body developments benefitted from exceptional support from ONR (Bobby Junker) and ARO (Mikal Ciftan), and their successors. But it is true that ALL these accomplishments are uniquely a research product of the DoD agencies who had the foresight to back them in their infancy. I am extremely appreciative of the confidence shown in our effort over these 40 years.