Multireference self-consistent-field energies without the many-electron wave function through a variational low-rank two-electron reduced-density-matrix method

The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17x10(9) determinants in the active space, could not be performed.     

Parametric two-electron reduced-density-matrix method applied to computing molecular energies and properties at nonequilibrium geometries

A parametric approach to the variational calculation of the two-electron reduced density matrix (2-RDM) for many-electron atoms and molecules has recently been developed in which the 2-RDM is parametrized to be both size consistent and nearly N-representable [C. Kollmar, J. Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. The parametric variational 2-RDM method is applied to computing ground-state molecular energies and properties at nonequilibrium geometries in significantly larger basis sets than previously employed. We study hydrogen abstraction from the hydroxide groups of H(2)O, NH(3)OH, and CH(3)OH. The 2-RDM method, parametrized by single and double excitations, shows significant improvement over coupled-cluster methods with similar excitations in predicting the shape of potential energy curves and bond-dissociation energies. Previous work completes the parametrization of the energy and 2-RDM by a system of n(2)h(2) normalization constraints, where n and h are the number of occupied and unoccupied orbitals, respectively. In the present paper, however, we show that the constraints can be eliminated by incorporating them into the energy and 2-RDM functions and, hence, the constrained optimization of the ground-state energy can be reformulated as an unconstrained optimization. The 2-RDMs from the parametric method are very nearly N-representable, and as measured by an l(2) norm, they are more accurate than the 2-RDMs from configuration interaction truncated at single and double excitations by an order of magnitude.     

Photoexcited tautomerization of vinyl alcohol to acetylaldehyde via a conical intersection from contracted Schrödinger theory

The photoexcited tautomerization of vinyl alcohol to acetylaldehyde via a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schr?dinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A, 2009, 80, 022507]. We computed absolute energies of the critical points as well as various intermediate points along the ground- and excited-state potential energy surface of vinyl alcohol and acetylaldehyde. The ACSE, MCSCF, second-order multireference many-body perturbation theory (MRMP2), and various coupled cluster methods all demonstrate the existence of a family of pathways from vinyl alcohol to acetylaldehyde via a conical intersection that are monotonically decreasing in energy. The conical intersection, proposed for the first time in this paper, is both structurally and energetically similar to the ground-state transition state. We observe a relationship between conical intersections and transition states both in this paper and in our previous work on bicyclobutane's ring conical intersection [J. W. Snyder, Jr. and D. A. Mazziotti, J. Chem. Phys., 2011, 135, 024107]. To treat multireference correlation, we seeded the ACSE with an initial 2-RDM guess from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE recovers more single-reference correlation energy than MRMP2 and more multireference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.     

Molecular properties from variational reduced-density-matrix theory with three-particle N-representability conditions

Molecular ground-state energies and two-electron reduced density matrices (2-RDMs) have recently been computed without the many-electron wave function by constraining the 2-RDM to satisfy a complete set of three-positivity conditions for N representability [D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006)]. Energies at both equilibrium and nonequilibrium geometries are obtained within 0.3% of the correlation energy. In this paper the authors extend this work to examine the accuracy of molecular properties, including multipole moments and components of the ground-state energy, relative to full configuration interaction (FCI). Comparisons are also made with 2-RDM methods with two-positivity conditions and two-positivity plus the generalized T1T2 conditions as well as several approximate wave function methods. Using the 2-RDM method with three-positivity conditions, the authors obtain dipole, quadrupole, and octupole moments for BeH2, BH, H2O, CO, and NH3 at equilibrium geometries that are within 0.04% of their FCI values. In addition, for the potential energy surface of N2, the 2-RDM method with three-positivity yields not only accurate total ground-state energies but also accurate expectation values of the kinetic energy operator, the electron-nuclei potential, and electron-electron repulsion.     

Two-electron reduced density matrices from the anti-Hermitian contracted Schrodinger equation: enhanced energies and properties with larger basis sets

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.     

Strongly correlated mechanisms of a photoexcited radical reaction from the anti-Hermitian contracted Schrödinger equation

Photoexcited radical reactions are critical to processes in both nature and materials, and yet they can be challenging for electronic structure methods due to the presence of strong electron correlation. Reduced-density-matrix (RDM) methods, based on solving the anti-Hermitian contracted Schro?dinger equation (ACSE) for the two-electron RDM (2-RDM), are examined for studying the strongly correlated mechanisms of these reactions with application to the electrocyclic interconversion of allyl and cyclopropyl radicals. We combine recent extensions of the ACSE to excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary spin states [A. E. Rothman, J. J. Foley IV, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. The ACSE predicts that the ground-state ring closure of the allyl radical has a high 52.5 kcal/mol activation energy that is consistent with experimental data, while the closure of an excited allyl radical can occur by disrotatory and conrotatory pathways whose transition states are essentially barrierless. Comparisons are made with multireference second- and third-order perturbation theories and multireference configuration interaction. While predicted energy differences do not vary greatly between methods, the ACSE appears to improve these differences when they involve a strongly and a weakly correlated radical by capturing a greater share of single-reference correlation that increases the stability of the weakly correlated radicals. For example, the ACSE predicts a -39.6 kcal/mol conversion of the excited allyl radical to the ground-state cyclopropyl radical in comparison to the -32.6 to -37.3 kcal/mol conversions predicted by multireference methods. In addition, the ACSE reduces the computational scaling with the number of strongly correlated orbitals from exponential (traditional multireference methods) to quadratic. Computed ground- and excited-state 2-RDMs are nearly N-representable.     

Testing the parametric two-electron reduced-density-matrix method with improved functionals: application to the conversion of hydrogen peroxide to oxywater

Parametrization of the two-electron reduced density matrix (2-RDM) has recently enabled the direct calculation of electronic energies and 2-RDMs at the computational cost of configuration interaction with single and double excitations. While the original Kollmar energy functional yields energies slightly better than those from coupled cluster with single-double excitations, a general family of energy functionals has recently been developed whose energies approach those from coupled cluster with triple excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. In this paper we test the parametric 2-RDM method with one of these improved functionals through its application to the conversion of hydrogen peroxide to oxywater. Previous work has predicted the barrier from oxywater to hydrogen peroxide with zero-point energy correction to be 3.3-to-3.9 kcal/mol from coupled cluster with perturbative triple excitations [CCSD(T)] and -2.3 kcal/mol from complete active-space second-order perturbation theory (CASPT2) in augmented polarized triple-zeta basis sets. Using a larger basis set than previously employed for this reaction-an augmented polarized quadruple-zeta basis set (aug-cc-pVQZ)-with extrapolation to the complete basis-set limit, we examined the barrier with two parametric 2-RDM methods and three coupled cluster methods. In the basis-set limit the M parametric 2-RDM method predicts an activation energy of 2.1 kcal/mol while the CCSD(T) barrier becomes 4.2 kcal/mol. The dissociation energy of hydrogen peroxide to hydroxyl radicals is also compared to the activation energy for oxywater formation. We report energies, optimal geometries, dipole moments, and natural occupation numbers. Computed 2-RDMs nearly satisfy necessary N-representability conditions.     

Cumulant reconstruction of the three-electron reduced density matrix in the anti-Hermitian contracted Schrödinger equation

Differing perspectives on the accuracy of three-electron reduced-density-matrix (3-RDM) reconstruction in nonminimal basis sets exist in the literature. This paper demonstrates the accuracy of cumulant-based reconstructions, developed by Valdemoro (V) [F. Colmenero et al., Phys. Rev. A 47, 971 (1993)], Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996)], Mazziotti (M) [Phys. Rev. A 60, 3618 (1999)], and Valdemoro-Tel-Perez-Romero (VTP) [Many-electron Densities and Density Matrices, edited by J. Cioslowski (Kluwer, Boston, 2000)]. Computationally, we extend previous investigations to study a variety of molecules, including LiH, HF, NH(3), H(2)O, and N(2), in Slater-type, double-zeta, and polarized double-zeta basis sets at both equilibrium and nonequilibrium geometries. The reconstructed 3-RDMs, compared with 3-RDMs from full configuration interaction, demonstrate in nonminimal basis sets the accuracy of the first-order expansion (V) as well as the important role of the second-order corrections (NY, M, and VTP). Calculations at nonequilibrium geometries further show that cumulant functionals can reconstruct the 3-RDM from a multireferenced 2-RDM with reasonable accuracy, which is relevant to recent multireferenced formulations of the anti-Hermitian contracted Schrodinger equation (ACSE) and canonical diagonalization. Theoretically, we perform a detailed perturbative analysis of the M functional to identify its second-order components. With these second-order components we connect the M, NY, and VTP reconstructions for the first time by deriving both the NY and VTP functionals from the M functional. Finally, these 3-RDM reconstructions are employed within the ACSE [D. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)] to compute ground-state energies which are compared with the energies from the contracted Schrodinger equation and several wave function methods.     

Balancing single- and multi-reference correlation in the chemiluminescent reaction of dioxetanone using the anti-Hermitian contracted Schrödinger equation

Direct computation of energies and two-electron reduced density matrices (2-RDMs) from the anti-Hermitian contracted Schro?dinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)], it is shown, recovers both single- and multi-reference electron correlation in the chemiluminescent reaction of dioxetanone especially in the vicinity of the conical intersection where strong correlation is important. Dioxetanone, the light-producing moiety of firefly luciferin, efficiently converts chemical energy into light by accessing its excited-state surface via a conical intersection. Our previous active-space 2-RDM study of dioxetanone [L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010)] concluded that correlating 16 electrons in 13 (active) orbitals is required for realistic surfaces without correlating the remaining (inactive) orbitals. In this paper we pursue two complementary goals: (i) to correlate the inactive orbitals in 2-RDMs along dioxetanone's reaction coordinate and compare these results with those from multireference second-order perturbation theory (MRPT2) and (ii) to assess the size of the active space-the number of correlated electrons and orbitals-required by both MRPT2 and ACSE for accurate energies and surfaces. While MRPT2 recovers very different amounts of correlation with (4,4) and (16,13) active spaces, the ACSE obtains a similar amount of correlation energy with either active space. Nevertheless, subtle differences in excitation energies near the conical intersection suggest that the (16,13) active space is necessary to determine both energetic details and properties. Strong electron correlation is further assessed through several RDM-based metrics including (i) total and relative energies, (ii) the von Neumann entropy based on the 1-electron RDM, as well as the (iii) infinity and (iv) squared Frobenius norms based on the cumulant 2-RDM.     

Treating molecules in arbitrary spin states using the parametric two-electron reduced-density-matrix method

Minimizing the electronic energy with respect to a parameterized two-electron reduced density matrix (2-RDM) is known as a parametric variational 2-RDM method. The parametric 2-RDM method with the M 2-RDM parametrization [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] is extended to treat molecules in arbitrary spin states. Like its singlet counterpart, the M parametric 2-RDM method for arbitrary spin states is derived using approximate N-representability conditions, which allow it to capture more correlation energy than coupled cluster with single and double excitations at a lower computational cost. We present energies, optimized bond lengths, potential energy curves, and occupation numbers for a set of molecules in a variety of spin states using the M and K parametric 2-RDM methods as well as several wavefunction methods. We show that the M parametric 2-RDM method can describe bond breaking of open-shell molecules like triplet B(2) and singlet and triplet OH(+) even in the presence of strong correlation. Finally, the computed 2-RDMs are shown to be nearly N-representable at both equilibrium and non-equilibrium geometries.     

Conical intersection of the ground and first excited states of water: energies and reduced density matrices from the anti-Hermitian contracted Schrödinger equation

A conical intersection between the ground and first-excited states of water is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schr?dinger equation (ACSE). This study is an extension of a previous study in which the ACSE was used to compute the energies around a conical intersection in the triplet excited states of methylene [Snyder, J. W., Jr.; Rothman, A. E.; Foley, J. J.; Mazziotti, D. A. J. Chem. Phys. 2010, 132, 154109]. We compute absolute energies of the 1(1)A' and 2(1)A' states of water (H(2)O) and the location of the conical intersection. The ACSE energies are compared to those from ab initio wave function methods. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. Unlike the situation for methylene, the two states in the vicinity of the conical intersection of water both have the same spatial symmetry. Hence, the study demonstrates the ability of the ACSE to resolve states of the same spatial symmetry that are nearly degenerate in energy. The 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from double-ζ and augmented double-ζ basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.     

First singlet (n,pi*) excited state of hydrogen-bonded complexes between water and pyrimidine

Hydrogen bonds from water to excited-state formaldehyde and from water to excited-state pyridine have been shown to display novel motifs to traditional hydrogen bonds involving ground states, with, in particular for H2O:pyridine, strong interactions involving the electron-rich pi cloud dominating the (n,pi) excited state. We investigate H2O:pyrimidine and various dihydrated species and reveal another motif, one in which the hydrogen bonding can dramatically alter the electronic structure of the excited state. Such effects are rare for ground-state interactions for which hydrogen bonding usually acts to merely perturb the electronic structure of the participating molecules. It arises as the (n,pi*) excitation of isolated pyrimidine is delocalized over both nitrogens but asymmetric hydrogen bonding causes it to localize on just the noninteracting atom. As a result, the excited-state hydrogen bond in H2O:pyrimidine is suprisingly very similar to the ground-state structure. These results lead to an improved understanding of the spectroscopy of pyrimidine in liquid water, and to the prediction that stable excited-state hydrogen bonds in H2O:pyrimidine should be observable, despite failure of experiments to actually do so. They also provide a simple model for the intricate control over primary charge separation in photosynthesis exerted by hydrogen bonding, and for solvent-induced electron localization in symmetric mixed-valence complexes. All conclusions are based on strong parallels found between the results of calculations performed using density-functional theory (DFT) and time-dependent DFT (TDDFT), complete-active-space self-consistent-field (CASSCF) with second-order perturbation-theory correction (CASPT2) theory, and equation-of-motion coupled cluster (EOM-CCSD) theory, calculations that are verified through detailed comparison of computed properties with experimental data for both the isolated molecules and the ground-state hydrogen bond.     

How accurate are TD-DFT excited-state geometries compared to DFT ground-state geometries?

In this work, we take a different angle to the benchmarking of time-dependent density functional theory (TD-DFT) for the calculation of excited-state geometries by extensively assessing how accurate such geometries are compared to ground-state geometries calculated with ordinary DFT. To this end, we consider 20 medium-sized aromatic organic compounds whose lowest singlet excited states are ideally suited for TD-DFT modeling and are very well described by the approximate coupled-cluster singles and doubles (CC2) method, and then use this method and six different density functionals (BP86, B3LYP, PBE0, M06-2X, CAM-B3LYP, and ωB97XD) to optimize the corresponding ground- and excited-state geometries. The results show that although each hybrid functional reproduces the CC2 excited-state bond lengths very satisfactorily, achieving an overall root mean square error of 0.011 Å for all 336 bonds in the 20 molecules, these errors are distinctly larger than those of only 0.004–0.006 Å with which the hybrid functionals reproduce the CC2 ground-state bond lengths. Furthermore, for each functional employed, the variation in the error relative to CC2 between different molecules is found to be much larger (by at least a factor of 3) for the excited-state geometries than for the ground-state geometries, despite the fact that the molecules/states under investigation have rather uniform chemical and spectroscopic character. Overall, the study finds that even in favorable circumstances, TD-DFT excited-state geometries appear intrinsically and comparatively less accurate than DFT ground-state ones.     

Time-dependent Density Functional Theory Study on the Electronically Excited States of N-Methylformamide in Aqueous Solution

Excited-state hydrogen-bonding dynamics of N-methylformamide (NMF) in water has been investigated by time-dependent density functional theory (TDDFT) method. The ground-state geometry optimizations were calculated by density functional theory (DFT) method, while the electronic transition energies and corresponding oscillation strengths of the low-lying electronically excited states of isolated NMF, water monomers and the hydrogen-bonded NMF-H 2 O were calculated by TDDFT method. According to Zhao's rule on the excited-state hydrogen bonding dynamics, our results demonstrate that the intermolecular hydrogen bond C=O···O-H is strengthened and weakened in different electronically excited states. The hydrogen bond strengthening and weakening in the electronically excited state plays an important role in the photophysics of NMF in solutions.     

A computational study of the ground and excited state structure and absorption spectra of free-base N-confused porphine and free-base N-confused tetraphenylporphyrin

Computational investigations into the ground and singlet excited-state structures and the experimental ground-state absorption spectra of N-confused tetraphenylporphyrin tautomers 1e and 1i and N-confused porphines (NCP) 2e and 2i have been performed. Structural data for the ground state, performed at the B3LYP/6-31G(d), B3LYP/6-31+G(d)//B3LYP/6-31G(d), and B3LYP/6-311+G(d)//B3LYP/6-31G(d) levels, are consistent with those performed at lower levels of theory. Calculations of the gas-phase, ground-state absorption spectrum are qualitatively consistent with condensed phase experiments for predicting the relative intensities of the Q(0,0) and Soret bands. Inclusion of implicit solvation in the calculations substantially improves the correlation of the energy of the Soret band with experiment for both tautomers (1e, 435 nm predicted, 442 nm observed in DMAc; 1i, 435 nm predicted, 437 nm observed in CH2Cl2). The x- and y-polarized Q-band transitions were qualitatively reproduced for 1e in both the gas phase and with solvation, although the low-energy absorption band in 1i was predicted at substantially higher energy (646 nm in the gas phase and 655 nm with solvation) than observed experimentally (724 nm in CH2Cl2). Franck-Condon state and equilibrated singlet excited-state geometries were calculated for unsubstituted NCP tautomers 2e and 2i at the TD-B3LYP/SVP and TD-B3LYP/TZVP//TD-B3LYP/SVP levels. Electronic difference density plots were calculated from these geometries, thereby indicating the change of electron density in the singlet excited states. Adiabatic S1 and S2 geometries of these compounds were also calculated at the TD-B3LYP/SVP level, and the results indicate that while 2i is a more stable ground-state molecule by approximately 7.0 kcal mol-1, the energy difference for the S1 excited states is only approximately 1.0 kcal mol-1 and is 6.1 kcal mol-1 for the S2 excited states.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

Characterization of the methoxy carbonyl radical formed via photolysis of methyl chloroformate at 193.3 nm

This study investigates two features of interest in recent work on the photolytic production of the methoxy carbonyl radical and its subsequent unimolecular dissociation channels. Earlier studies used methyl chloroformate as a photolytic precursor for the CH3OCO, methoxy carbonyl (or methoxy formyl) radical, which is an intermediate in many reactions that are relevant to combustion and atmospheric chemistry. That work evidenced two competing C-Cl bond fission channels, tentatively assigning them as producing ground- and excited-state methoxy carbonyl radicals. In this study, we measure the photofragment angular distributions for each C-Cl bond fission channel and the spin-orbit state of the Cl atoms produced. The data shows bond fission leading to the production of ground-state methoxy carbonyl radicals with a high kinetic energy release and an angular distribution characterized by an anisotropy parameter, beta, of between 0.37 and 0.64. The bond fission that leads to the production of excited-state radicals, with a low kinetic energy release, has an angular distribution best described by a negative anisotropy parameter. The very different angular distributions suggest that two different excited states of methyl chloroformate lead to the formation of ground- and excited-state methoxy carbonyl products. Moreover, with these measurements we were able to refine the product branching fractions to 82% of the C-Cl bond fission resulting in ground-state radicals and 18% resulting in excited-state radicals. The maximum kinetic energy release of 12 kcal/mol measured for the channel producing excited-state radicals suggests that the adiabatic excitation energy of the radical is less than or equal to 55 kcal/mol, which is lower than the 67.8 kcal/mol calculated by UCCSD(T) methods in this study. The low-lying excited states of methylchloroformate are also considered here to understand the observed angular distributions. Finally, the mechanism for the unimolecular dissociation of the methoxy carbonyl radical to CH3 + CO2, which can occur through a transition state with either cis or, with a much higher barrier, trans geometry, was investigated with natural bond orbital computations. The results suggest donation of electron density from the nonbonding C radical orbital to the sigma* orbital of the breaking C-O bond accounts for the additional stability of the cis transition state.     

First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules

Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.