# Research

Following the initial formulation of Jiri Cizek of coupled-cluster theory with double excitations, CCD, Rod Bartlett pioneered the development of coupled-cluster (CC) theory in quantum chemistry to offer highly accurate solutions of the Schrödinger equation for molecular structure and spectra. He introduced the term ‘size-extensivity’ for many-body methods like CC that scale properly with the number of electrons; now viewed as an essential element in quantum chemistry approximations.

He and his co-workers were the first to formulate and implement CC theory with all single and double excitations (CCSD), to add triples non-iteratively (CCSD[T]), iteratively, CCSDT-1, and fully, CCSDT; followed by quadruple (CCSDTQ) and pentuple excitations (CCSDTQP). He also presented a density matrix formulation for the analytical gradients (forces) for the non-variational CC method to identify structures, transition states, and vibrational spectra.; a necessity for any widely used method in quantum chemistry. This led to the CC functional, E = 〈0|(1+Λ)exp(-T)Hexp(T)|0〉=〈0|(1+Λ)Ħ|0〉 where the forces on an atom, a, become ∂E/∂Xa = 〈0|(1+Λ)exp(-T)(∂H/∂Xa)exp(T)|0〉.. This functional also defines the CC response and relaxed density matrices for all properties, providing a non-Hermitian generalization of the conventional density matrices that is applicable to methods even without a wavefunction, like CCSD(T).

Excited, ionized, etc. states in CC are obtained from the equation-of-motion CC, ωk =〈0|LkĦRk|0〉, where ψk=Rkexp(T)|0〉. With energies, analytical forces, excited states, density matrices, and other properties, CC theory has now been documented to offer the most predictive, widely applicable results in the field.

Thus, Bartlett’s group has been instrumental in establishing the now well-accepted paradigm MP2<CCD<CCSD<CCSD(T)<CCSDT<CCSDT(Qf)<CCSDTQ<Full CI, for converging, many-body, quantum chemical methods. The history is summarized in the article, ““How and Why Coupled-cluster Theory Become the Pre-eminent *Ab Initio *Method in Quantum Chemistry?” in *Theory and Applications of Computational Chemistry: The First Forty Years,* (C. Dykstra, G. Frenking, K. Kim and G. Scuseria, editors) Elsevier, 1191-1221 (2005).

**Other research topics include:**

The search for metastable, high-energy density molecules (HEDM) like N4 N8, and N5-, which he has long predicted to exist (The pentazole anion, an aromatic five-membered ring, has been recently observed for the first time in negative ion mass spectra verifying his prediction. Inconclusive observations of N4 based upon Bartlett’s group predicted vibrational frequencies have also been made. Also the analog, N3O+ has also been seen in mass spectra.

Non-linear optical properties of molecules, where his group’s work resolved long-standing discrepancies between theory and electric-field induced second and third harmonic generation experiments. The new theory produced in this work introduced any-order time-dependent Hartree-Fock theory for frequency dependent properties and that for the initial time-dependent CC results.

Carbon clusters, where his group’s work on the rhombic form of C4, which he found to be competitive in stability with its linear triplet form, has been instrumental in the closed-shell vs. open-shell debate about small carbon clusters. Cyclic forms of C5 and C6 have been observed spectroscopically, while reports of rhombic C4 have been reported in Coulomb explosion experiments.

NMR coupling constants His group’s EOM-CCSD work was the first to offer predictive results for NMR coupling constants whose average errors are ~ 3Hz. With this tool, he provided fingerprints for the non-classical bridged H atom in ethylcarbenium and the bridged, pentacoordinate C atom in the 2-norbornyl cation at the heart of the Olah and Brown debate about non-classical carbocations, which had resisted experimental determination. The observed coupling constants are also in exceptional agreement with those that could be obtained experimentally by Olah, substantiating the accuracy of Bartlett’s predictions. For H bonds his group provides formulae to relate the two-bond coupling constant to the distance between the atoms that are H-bonded which potentially provides a new probe to assist biomolecular structure determination that is complementary to X-ray determination where the H atoms cannot be observed.

**His group continually introduces new correlated quantum chemical methods:**

EOM-CC and STEOM-CC for excited, ionized, and electron attached states for molecules, recently generalizing the former to the full EOM-CCSDT. Like the CC methods for ground states, these methods offer the same unambiguous application to excited states, with similar successes.

New correlated methods for polymers, reporting the first CCSD results a few years ago.

*Ab Initio* density functional theory, an approach that unlike other current hybrid or gradient corrected DFT methods has to converge to the right answer in the limit like *ab initio* quantum chemistry. The most recent work derives the exact exchange-correlation potential of DFT from coupled-cluster theory, making a seamless connection between wave-function theory and density functional theory.

The ‘transfer Hamiltonian’ procedure to make it possible to do quantum mechanically based, ‘predictive’ simulations for materials.

The natural linear scaled coupled-cluster method (NLSCC) which unlike others, makes it possible to use large basis sets in the quasi-separable units based upon natural bond localized orbitals. It also has the advantage that it only uses orthogonal unoccupied molecular orbitals which has significant computational advantages over projected atomic orbital methods. NLSCC also provides transferable cluster amplitudes that reflect conventional chemical intuition. In [particular, this method provides and understanding of energetic molecules based upon transferable units like –CH2-NNO, and the others that are important in explosives.