Potential-Energy Curves for the Ground and Several Electronic States of NdO and NdS.

Potential energy curves (PECs) were calculated for 21 and 18 electronic states of NdO and NdS molecules, respectively. In each case, static electron correlation effects were described by incomplete model space multiconfiguration self-consistent field wave functions based on an active space that included the most important valence orbitals. Dynamic electron correlation was included by the multireference second-order generalized Van Vleck perturbation theory method. Scalar-relativistic contributions were included by the effective core potential approach, using def2-TZVPP basis sets. Spin-dependent relativistic corrections were determined to be small and negligible for the Nd atom and so were not included in the calculations. The 21 and 18 electronic states of NdO and NdS were predicted to be in the excitation energy range of ∼3.2 and ∼2.7 eV, respectively. The ground electronic states of NdO and NdS were determined as 15H (6s4fσ4fϕ4fδ) and 15H (4fϕ4fπ4fπ6s), with spectroscopic constants: bond length Re = 1.780 and 2.325 Å, and harmonic frequency ωe = 891 and 538 cm-1, respectively.

Microbe-dependent heterosis in maize.

Hybrids account for nearly all commercially planted varieties of maize and many other crop plants because crosses between inbred lines of these species produce first-generation [F1] offspring that greatly outperform their parents. The mechanisms underlying this phenomenon, called heterosis or hybrid vigor, are not well understood despite over a century of intensive research. The leading hypotheses-which focus on quantitative genetic mechanisms (dominance, overdominance, and epistasis) and molecular mechanisms (gene dosage and transcriptional regulation)-have been able to explain some but not all of the observed patterns of heterosis. Abiotic stressors are known to impact the expression of heterosis; however, the potential role of microbes in heterosis has largely been ignored. Here, we show that heterosis of root biomass and other traits in maize is strongly dependent on the belowground microbial environment. We found that, in some cases, inbred lines perform as well by these criteria as their F1 offspring under sterile conditions but that heterosis can be restored by inoculation with a simple community of seven bacterial strains. We observed the same pattern for seedlings inoculated with autoclaved versus live soil slurries in a growth chamber and for plants grown in steamed or fumigated versus untreated soil in the field. In a different field site, however, soil steaming increased rather than decreased heterosis, indicating that the direction of the effect depends on community composition, environment, or both. Together, our results demonstrate an ecological phenomenon whereby soil microbes differentially impact the early growth of inbred and hybrid maize.

Exploring the biointerfaces: ab initio investigation of nano-montmorillonite clay, and its interaction with unnatural amino acids.

We investigated the interaction between biomimetic Fe and Mg co-doped montmorillonite nanoclay and eleven unnatural amino acids. Employing three different functionals (PBE-GGA, PBE-GGA + U, and HSE06), we examined the clay's structural, electronic, and magnetic properties. Our results revealed the necessity of using PBE-GGA + U with U ≥ 4 eV to accurately describe key clay properties. We identified amino acids that strongly interacted with the clay surface, with steric orientation playing a crucial role in facilitating binding. Our DFT calculations highlighted significant electrostatic interactions between the amino acids and the clay slab, with the amino group's predominant role in this interaction. These findings hold promise for designing amino acids for clay-amino acid systems, leading to innovative bio-material composites for various applications. Additionally, our ab-initio molecular dynamics simulations confirmed the stability of clay-amino acid systems under ambient conditions, and the introduction of an implicit water solvent enhanced the binding energy of amino acids on the clay surface.

Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.

Theoretical Calculations of the 242 nm Absorption of Propargyl Radical.

The propargyl radical, the most stable isomer of neutral C3H3, is important in combustion reactions, and a number of spectroscopic and reaction dynamics studies have been performed over the years. However, theoretical calculations have never been able to find a state that can generate strong absorption around 242 nm as seen in experiments. In this study, we calculated the low-lying electronic energy levels of the propargyl radical using the highly accurate multireference configuration interaction singles and doubles method with triples and quadruples treated perturbatively [denoted as MRCISD(TQ)]. Calculations indicate that this absorption can be attributed to a Franck-Condon-allowed electronic transition from the ground 2B1 state to the Rydberg-like excited state 12A1. Further insight into the behavior of the multireference perturbative theory methods, GVVPT2 and GVVPT3, on a very challenging system are also obtained.

Mental Health and Health-Related Quality of Life Among Nephrology Nurses: A Survey-Based Cross-Sectional Study.

Nephrology nurses face health and wellness challenges due to significant work-related stressors. This survey, conducted online between July 24 and August 17, 2020, assessed the psychological well-being of nephrology nurses in the United States during the COVID-19 pandemic (n = 393). Respondents reported feeling burned out from work (62%), symptoms of anxiety (47% with Generalized Anxiety Disorder-7 [GAD-7] scores ≥ 5), and major depressive episodes (16% with Patient Health Questionnaire-2 [PHQ-2] scores ≥ 3). Fifty-six percent (56%) of survey respondents reported caring for COVID-19 patients, and 62% were somewhat or very worried about COVID-19. Factors, including high workload, age, race, and the COVID-19 pandemic, may partially explain the high proportion of nephrology nurses who reported symptoms of burnout, anxiety, and depression.

iVI: An iterative vector interaction method for large eigenvalue problems.

Based on the generic "static-dynamic-static" framework for strongly coupled basis vectors (Liu and Hoffman, Theor. Chem. Acc. 2014, 133, 1481), an iterative Vector Interaction (iVI) method is proposed for computing multiple exterior or interior eigenpairs of large symmetric/Hermitian matrices. Although it works with a fixed-dimensional search subspace, iVI can converge quickly and monotonically from above to the exact exterior/interior roots. The efficacy of iVI is demonstrated by taking both mathematical and physical matrices as examples. © 2017 Wiley Periodicals, Inc.

Accurate Dissociation of Chemical Bonds Using DFT-in-DFT Embedding Theory with External Orbital Orthogonality.

Our recent density functional theory (DFT)-in-DFT embedding protocol, which enforces intersubsystem (or external orbital) orthogonality, is used for the first time to investigate covalent bond dissociation and is shown to do so accurately. Full potential energy curves for the dissociation of a H-O bond in H2O and the C-C bond in H3C-CH3 have been constructed using the new embedding method, as have the challenging ionic bonds in LiH and LiF, and were found to match the reference Kohn-Sham (KS)-DFT curves to at least one part in 106. The added constraint of external orbital orthogonality allows for the formulation of an embedding protocol that does not rely on approximate kinetic energy functionals for the evaluation of the so-called nonadditive kinetic potential, does not introduce compensatory potentials, and does not require a total system calculation at any stage. The present work extends the demonstrated applicability of the external orthogonality variant of embedding theory by more than a factor of 2 to the interaction strength range of strong single bonds. In particular, it is demonstrated that homolytic cleavage of both covalent and ionic bonds into radicals can be accomplished.

Inter-Population Variability of Endosymbiont Densities in the Asian Citrus Psyllid (Diaphorina citri Kuwayama).

The Asian citrus psyllid (Diaphorina citri Kuwayama) is an insect pest capable of transmitting Candidatus Liberibacter asiaticus (CLas), the causal agent of citrus greening in North America. D. citri also harbors three endosymbionts, Wolbachia, Candidatus Carsonella ruddii, and Candidatus Profftella armatura, which may influence D. citri physiology and fitness. Although genomic researches on these bacteria have been conducted, much remains unclear regarding their ecology and inter-population variability in D. citri. The present work examined the densities of each endosymbiont in adult D. citri sampled from different populations using quantitative PCR. Under field conditions, the densities of all three endosymbionts positively correlated with each other, and they are associated with D. citri gender and locality. In addition, the infection density of CLas also varied across populations. Although an analysis pooling D. citri from different populations showed that CLas-infected individuals tended to have lower endosymbiont densities compared to uninfected individuals, the difference was not significant when the population was included as a factor in the analysis, suggesting that other population-specific factors may have stronger effects on endosymbiont densities. To determine whether there is a genetic basis to the density differences, endosymbiont densities between aged CLas-negative females of two D. citri populations reared under standardized laboratory conditions were compared. Results suggested that inter-population variability in Wolbachia infection density is associated with the genotypes of the endosymbiont or the host. Findings from this work could facilitate understanding of D. citri-bacterial associations that may benefit the development of approaches for managing citrus greening, such as prevention of CLas transmission.

Competitive excited-state single or double proton transfer mechanisms for bis-2,5-(2-benzoxazolyl)-hydroquinone and its derivatives.

The excited state intramolecular proton transfer (ESIPT) mechanisms of 2-(2-hydroxyphenyl)benzoxazole (HBO), bis-2,5-(2-benzoxazolyl)-hydroquinone (BBHQ) and 2,5-bis(5'-tert-butyl-benzoxazol-2'-yl)hydroquinone (DHBO) have been investigated using time-dependent density functional theory (TDDFT). The calculated vertical excitation energies based on the TDDFT method reproduced the experimental absorption and emission spectra well. Three kinds of stable structures were found on the S1 state potential energy surface (PES). A new ESIPT mechanism that differs from the one proposed previously (Mordzinski et al., Chem. Phys. Lett., 1983, 101, 291. and Lim et al., J. Am. Chem. Soc., 2006, 128, 14542.) is proposed. The new mechanism includes the possibility of simultaneous double proton transfer, or successive single transfers, in addition to the accepted single proton transfer mechanism. Hydrogen bond strengthening in the excited state was based on primary bond lengths, angles, IR vibrational spectra and hydrogen bond energy. Intramolecular charge transfer based on the frontier molecular orbitals (MOs) also supports the proposed mechanism of the ESIPT reaction. To further elucidate the proposed mechanism, reduced dimensionality PESs of the S0 and S1 states were constructed by keeping the O-H distance fixed at a series of values. The potential barrier heights among the local minima on the S1 surface imply competitive single and double proton transfer branches in the mechanism. Based on the new ESIPT mechanism, the observed fluorescence quenching can be satisfactorily explained.

Use of density functional theory orbitals in the GVVPT2 variant of second-order multistate multireference perturbation theory.

A new variation of the second-order generalized van Vleck perturbation theory (GVVPT2) for molecular electronic structure is suggested. In contrast to the established procedure, in which CASSCF or MCSCF orbitals are first obtained and subsequently used to define a many-electron model (or reference) space, the use of an orbital space obtained from the local density approximation (LDA) variant of density functional theory is considered. Through a final, noniterative diagonalization of an average Fock matrix within orbital subspaces, quasicanonical orbitals that are otherwise indistinguishable from quasicanonical orbitals obtained from a CASSCF or MCSCF calculation are obtained. Consequently, all advantages of the GVVPT2 method are retained, including use of macroconfigurations to define incomplete active spaces and rigorous avoidance of intruder states. The suggested variant is vetted on three well-known model problems: the symmetric stretching of the O-H bonds in water, the dissociation of N2, and the stretching of ground and excited states C2 to more than twice the equilibrium bond length of the ground state. It is observed that the LDA-based GVVPT2 calculations yield good results, of comparable quality to conventional CASSCF-based calculations. This is true even for the C2 model problem, in which the orbital space for each state was defined by the LDA orbitals. These results suggest that GVVPT2 can be applied to much larger problems than previously accessible.

New excited-state proton transfer mechanisms for 1,8-dihydroxydibenzo[a,h]phenazine.

The excited state intramolecular proton transfer (ESIPT) mechanisms of 1,8-dihydroxydibenzo[a,h]phenazine (DHBP) in toluene solvent have been investigated based on time-dependent density functional theory (TD-DFT). The results suggest that both a single and double proton transfer mechanisms are relevant, in constrast to the prediction of a single one proposed previously (Piechowska et al. J. Phys. Chem. A 2014, 118, 144-151). The calculated results show that the intramolecular hydrogen bonds were formed in the S0 state, and upon excitation, the intramolecular hydrogen bonds between -OH group and pyridine-type nitrogen atom would be strengthened in the S1 state, which can facilitate the proton transfer process effectively. The calculated vertical excitation energies in the S0 and S1 states reproduce the experimental UV-vis absorption and fluorescence spectra well. The constructed potential energy surfaces of the S0 and S1 states have been used to explain the proton transfer process. Four minima have been found on the S1 state surface, with potential barriers between these excited-state minima of less than 10 kcal/mol, which supports concomitant single and double proton transfer mechanisms. In addition, the fluorescence quenching can be explained reasonably based on the proton transfer process.

Density differences in embedding theory with external orbital orthogonality.

First results on electron densities and energies for a number of molecular complexes with different interaction strengths (ranging from ca. 0.3 to 40 kcal/mol), obtained using our recently introduced DFT-in-DFT embedding equations (i.e., Kohn-Sham equations with constrained electron density (KSCED) and external orbital orthogonality (ext orth), KSCED(x, ext orth), where x denotes the single particle support: monomer (m); supermolecular (s); or extended monomer (e)) are compared with densities from supermolecular Kohn-Sham (KS)-DFT calculations and traditional DFT-in-DFT results. Because our methodology does not rely on error-prone potentials that are not present in supermolecular KS-DFT calculations, it allows DFT-in-DFT calculations to achieve much higher accuracy than previous protocols of DFT-in-DFT that employed such potentials. It is shown that whereas conventional DFT-in-DFT embedding theory leads to errors in the electron density at the boundary between subsystems, the situation is remedied when orbital orthogonality between subsystems (i.e., external orthogonality) is enforced. Our approach reproduces KS-DFT total energies at least to the seventh decimal place (and exactly at most geometries) for the tested systems. Potential energy curves (PECs) of the separation of some of the tested systems into fragments are calculated. PECs, obtained with the new equations, using the usual Kohn-Sham equations with constrained electron density and supermolecular basis expansion [KSCED(s, ext orth, v(T) = 0), where v(T) is the nonadditive kinetic potential] were found to be virtually identical to those from conventional KS-DFT; equilibrium distances and interaction energies were reproduced to all reported digits for both local density approximation (LDA) and generalized gradient approximation (GGA) functionals. As an additional approximation, an alternative one-particle space (to the common monomer or supermolecular spaces) in which KS orbitals of a subsystem are expanded is introduced. This expansion, which we refer to as the extended monomer expansion [e.g., KSCED(e)], includes basis functions centered on atom(s) of the complementary subsystem in the interfacial region. Density differences and PECs obtained with the new equations and new one-particle space [i.e., KSCED(e, ext orth, v(T) = 0)] were closely related to those obtained from KSCED(s, ext orth, v(T) = 0). The new approach does not require any supermolecular calculations.

Relativistic GVVPT2 multireference perturbation theory description of the electronic states of Y2 and Tc2.

The multireference generalized Van Vleck second-order perturbation theory (GVVPT2) method is used to describe full potential energy curves (PECs) of low-lying states of second-row transition metal dimers Y(2) and Tc(2), with scalar relativity included via the spin-free exact two-component (sf-X2C) Hamiltonian. Chemically motivated incomplete model spaces, of the style previously shown to describe complicated first-row transition metal diatoms well, were used and again shown to be effective. The studied states include the previously uncharacterized 2(1)Σ(g)(+) and 3(1)Σ(g)(+) PECs of Y(2). These states, together with 1(1)Σ(g)(+), are relevant to discussion of controversial results in the literature that suggest dissociation asymptotes that violate the noncrossing rule. The ground state of Y(2) was found to be X(5)Σ(u)(­) (similar to Sc(2)) with bond length R(e) = 2.80 Å, binding energy D(e) = 3.12 eV, and harmonic frequency ω(e) = 287.2 cm(­1), whereas the lowest 1(1)(g)(+) state of Y(2) was found to lie 0.67 eV above the quintet ground state and had spectroscopic constants R(e) = 3.21 Å, D(e) = 0.91 eV, and ω(e) = 140.0 cm(­1). Calculations performed on Tc(2) include study of the previously uncharacterized relatively low-lying 1(5)Σ(g)(+) and 1(9)Σ(g)(+) states (i.e., 0.70 and 1.84 eV above 1(1)Σ(g)(+), respectively). The ground state of Tc(2) was found to be X(3)Σ(g)(­) with R(e) = 2.13 Å, D(e) = 3.50 eV, and ω(e) = 336.6 cm(­1) (for the most stable isotope, Tc-98) whereas the lowest (1)Σ(g)(+) state, generally accepted to be the ground state symmetry for isovalent Mn(2) and Re(2), was found to lie 0.47 eV above the X(3)Σ(g)(­) state of Tc(2). The results broaden the range of demonstrated applicability of the GVVPT2 method.

The dissociation of glycolate-astrochemical and prebiotic relevance.

On the basis of mass spectrometric experiments and quantum chemical calculations, including detailed kinetic and dynamics calculations, we report the unimolecular dissociation of an isolated glycolate anion. The dominating processes are: loss of formaldehyde; loss of carbon monoxide; loss of carbon dioxide; and loss of a hydrogen molecule, with the latter having the lowest energetic threshold. At higher energies, CO loss is the dominating reaction. The loss of CO may be followed by a second CO loss, leading to the H(-)H2O complex in close mechanistic relationship to the Nibbering reaction. The results provide valuable insights into possible mechanisms for interstellar and prebiotic formation of glycolate via the reverse of the unimolecular dissociation reactions. We propose that the addition of the complex of OH(-) and CO to CH2O is the most feasible route to gas phase synthesis of glycolate, since all species are abundant in interstellar space.

Synthesis, structural characterization, photophysics, and broadband nonlinear absorption of a platinum(II) complex with the 6-(7-benzothiazol-2'-yl-9,9-diethyl-9 H-fluoren-2-yl)-2,2'-bipyridinyl ligand.

A platinum complex with the 6-(7-benzothiazol-2'-yl-9,9-diethyl-9H-fluoren-2-yl)-2,2'-bipyridinyl ligand (1) was synthesized and the crystal structure was determined. UV/Vis absorption, emission, and transient difference absorption of 1 were systematically investigated. DFT calculations were carried out on 1 to characterize the electronic ground state and aid in the understanding of the nature of low-lying excited electronic states. Complex 1 exhibits intense structured (1)π-π* absorption at λ(abs)<440 nm, and a broad, moderate (1)MLCT/(1)LLCT transition at 440-520 nm in CH(2)Cl(2) solution. A structured (3)π-π*/(3)MLCT emission at about 590 nm was observed at room temperature and at 77 K. Complex 1 exhibits both singlet and triplet excited-state absorption from 450 nm to 750 nm, which are tentatively attributed to the (1)π-π* and (3)π-π* excited states of the 6-(7-benzothiazol-2'-yl-9,9-diethyl-9H-fluoren-2-yl)-2,2'-bipyridine ligand, respectively. Z-scan experiments were conducted by using ns and ps pulses at 532 nm, and ps pulses at a variety of visible and near-IR wavelengths. The experimental data were fitted by a five-level model by using the excited-state parameters obtained from the photophysical study to deduce the effective singlet and triplet excited-state absorption cross sections in the visible spectral region and the effective two-photon absorption cross sections in the near-IR region. Our results demonstrate that 1 possesses large ratios of excited-state absorption cross sections relative to that of the ground-state in the visible spectral region; this results in a remarkable degree of reverse saturable absorption from 1 in CH(2)Cl(2) solution illuminated by ns laser pulses at 532 nm. The two-photon absorption cross sections in the near-IR region for 1 are among the largest values reported for platinum complexes. Therefore, 1 is an excellent, broadband, nonlinear absorbing material that exhibits strong reverse saturable absorption in the visible spectral region and large two-photon-assisted excited-state absorption in the near-IR region.

GVVPT2 multireference perturbation theory description of diatomic scandium, chromium, and manganese.

With relatively simple model spaces derived from valence bond models, a straightforward zero-order Hamiltonian, and the use of moderate-sized Dunning-type correlation consistent basis sets (cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ), the second order generalized Van Vleck perturbation theory (GVVPT2) method is shown to produce potential energy curves (PECs) and spectroscopic constants close to experimental results for both ground and low-lying excited electronic states of Sc(2), Cr(2) and Mn(2). In spite of multiple quasidegeneracies (particularly for the cases of Sc(2) and Mn(2)), the GVVPT2 PECs are smooth with no discontinuities. Since these molecules have been identified as ones that widely used perturbative methods are inadequate for describing well, due to intruder state problems, unless shift parameters are introduced that can obfuscate the physics, this study suggests that the conclusion about the inadequacy of multireference perturbation theory be re-evaluated. The ground state of Sc(2) is predicted to be X(5)∑(u)(-), and its spectroscopic constants are close to the ones at the MRCISD level. Near equilibrium geometries, the 1(3)∑(u)(-) electronic state of Sc(2) is found to be less stable than the quintet ground state by 0.23 eV. The Cr(2) PEC has several features of the Rydberg-Klein-Rees (RKR) experimental curve (e.g., the pronounced shelf at elongated bond lengths), although the predicted bond length is slightly long (R(e) = 1.80 Å with cc-pVQZ compared to the experimental value of 1.68 Å). The X(1)∑(g)(+) ground state of Mn(2) is predicted to be a van der Waals molecule with a long bond length, R(e), of 3.83 Å using a cc-pVQZ basis set (experimental value = 3.40 Å) and a binding energy, D(e), of only 0.05 eV (experimental value = 0.1 eV). We obtained R(e) = 3.40 Å and D(e) = 0.09 eV at the complete basis set (CBS) limit for ground state Mn(2). Low lying excited state curves have also been characterized for all three cases (Cr(2), Mn(2), and Sc(2)) and show similar mathematical robustness as the ground states. These results suggest that the GVVPT2 multireference perturbation theory method is more broadly applicable than previously documented.

Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: theory and test calculations of second order approximation.

A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo̸ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4, and BeH(2), are performed in order to test the new methods on problems where full configuration interaction results are available.

GVVPT2 energy gradient using a Lagrangian formulation.

A Lagrangian based approach was used to obtain analytic formulas for GVVPT2 energy nuclear gradients. The formalism can use either complete or incomplete model (or reference) spaces, and is limited, in this regard, only by the capabilities of the MCSCF program. An efficient means of evaluating the gradient equations is described. Demonstrative calculations were performed and compared with finite difference calculations on several molecules and show that the GVVPT2 gradients are accurate. Of particular interest, the suggested formalism can straightforwardly use state-averaged MCSCF descriptions of the reference space in which the states have arbitrary weights. This capability is demonstrated by some calculations on the ground and first excited singlet states of LiH, including calculations near an avoided crossing. The accuracy and usefulness of the GVVPT2 method and its gradient are highlighted by comparing the geometry of the near-C(2v) minimum on the conical intersection seam between the 1 (1)A(1) and 2 (1)A(1) surfaces of O(3) with values that were calculated at the multireference configuration interaction, including single and double excitations (MRCISD), level of theory.