Complete Active Space Perturbation Theory Research Articles

Ab Initio Calculation of UV-vis Absorption of Parent Mg, Fe, Co, Ni, Cu, and Zn Metalloporphyrins.

Relativistic restricted active space (RAS) second-order multireference perturbation theory (MRPT2) methods, incorporating spin-orbit (SO) coupling perturbatively via state interaction (SO-MRPT2/RASSCF), were used to reproduce the absorption spectra of parent metalloporphyrins containing the Mg2+, Zn2+, Co2+, Ni2+, Cu2+, or FeCl2+ ions in the 12,500-40,000 cm-1 region. Particular attention was paid to the interaction between the porphyrin ring and the metal 3d electrons in states of different multiplicities (we used metal 3d and double d-shell or 3d' orbitals). For this class of compounds, the N-electron valence state perturbation theory (NEVPT2) method is superior to the complete active space perturbation theory (CASPT2) and successfully reproduces the energies of all four characteristic transitions (Q, B, N, and L) of closed-shell metalloporphyrins. Inclusion of SO coupling was found to have very little effect on excitation energies and oscillator strengths. For FeCl2+ porphyrin, we treated ligand-to-metal charge-transfer (LMCT; π,d), metal ligand field (d,d), and metal-to-ligand charge-transfer (MLCT; d,π*) transitions within the same framework. The broad and intense spectral features associated with its B (Soret) band are attributed to multiconfigurational LMCT (d,π*) bands involving strong metal-ligand orbital mixing.

Excited-state van der Waals potential energy surfaces for the NO A2Σ+ + CO2X1Σg+ collision complex.

Excited state van der Waals (vdW) potential energy surfaces (PESs) of the NO A2Σ+ + CO2X1Σg+ system are thoroughly investigated using coupled cluster theory and complete active space perturbation theory to second order (CASPT2). First, it is shown that pair natural orbital coupled cluster singles and doubles with perturbative triples yields comparable accuracy compared to CCSD(T) for molecular properties and vdW-minima at a fraction of computational cost of the latter. Using this method in conjunction with highly diffuse basis sets and counterpoise correction for basis set superposition error, the PESs for different intermolecular orientations are investigated. These show numerous vdW-wells, interconnected for all geometries except one, with a maximum depth of up to 830 cm-1; considerably deeper than those on the ground state surface. Multi-reference effects are investigated with CASPT2 calculations. The long-range vdW-surfaces support recent experimental observations relating to rotational energy transfer due the anisotropy in the potentials.

An embedded cluster CASPT2 study of the Ce:YVO4 spectrum.

Multiconfigurational theory, in combination with the embedded cluster approach, is a precise and abinitio approach to describe the electronic structure of solids. In this work, the spectrum of a Ce(III) dopant in YVO4 has been studied by complete active space perturbation theory of the second order (CASPT2), with the host material represented as a set of abinitio model potentials and point-charges. We assess the sensitivity of the spectrum to the size of both the embedded cluster size as well as the size of the electronic basis set. A comparison of our best computational model with experimental results shows that the embedding approach is robust and can accurately model the spectrum of low-concentration dopants in complex host materials.

Effect of geometrical and electronic alteration on the n → π* vertical excitation energy and thermodynamic stability of locked azobenzene (LAB) with protein confinement

AbstractLocked azobenzenes (LAB) derivatives are recognized as photoswitchable potassium ion channel blockers. Here we report the effect of tetraethyl ammonium ion (TEA) substituent and KcsA K+‐ion channel protein environment on the thermodynamic stability and vertical excitation energy of locked azobenzene. The complete active space self‐consistent field (CASSCF and state‐averaged CASSCF, SA2‐CAS) with complete active space perturbation theory (state‐averaged CASPT2, SA2‐CASPT2) and our own n‐layered integrated molecular orbital and molecular mechanics—mechanical embedding (ONIOM‐ME) followed by electronic embedding (ONIOM‐EE) methods were employed for the study. The Z‐isomer of LAB is thermodynamically more stable as compared to E‐isomer. The substitution of tetraethyl ammonium ion does not influence the thermodynamic stability. On the other hand, the thermodynamic stability is reversed in the protein environment. Furthermore, the vertical excitation energies (VEE) and geometrical parameters show that the photochemistry of LAB is unaffected by para‐substituted TEA. However, geometrical changes imposed by the protein on the confined LAB‐TEA decreases the VEE.

Photoionization Observables from Multi-Reference Dyson Orbitals Coupled to B-Spline DFT and TD-DFT Continuum.

We present a theoretical model to compute the accurate photoionization dynamical parameters (cross-sections, asymmetry parameters and orbital, or cross-section, ratios) from Dyson orbitals obtained with the multi-state complete active space perturbation theory to the second order (MS-CASPT2) method. Our new implementation of Dyson orbitals in OpenMolcas takes advantage of the full Abelian symmetry point group and has the corrected normalization. The Dyson orbitals are coupled to an accurate description of the electronic continuum obtained with a multicentric B-spline basis at the DFT and TD-DFT levels. Two prototype diatomic molecules, i.e., CS and SiS, have been chosen due to their smallness, which hides important correlation effects. These effects manifest themselves in the appearance of well-characterized isolated satellite bands in the middle of the valence region. The rich satellite structures make CS and SiS the perfect candidates for a computational study based on our highly accurate MS-CASPT2/B-spline TD-DFT protocol.

QM/MM Investigation of the Spectroscopic Properties of the Fluorophore of Bacterial Luciferase.

We employ replica-exchange molecular dynamics (REMD) and a hybrid ab initio multiconfigurational quantum mechanics/molecular mechanics (QM/MM) approach to model the absorption and fluorescence properties of bacterial luciferin-luciferase. Specifically, we employ complete active space perturbation theory (CASPT2) and study the effect of active space, basis set, and IPEA shift on the computed energies. We discuss the effect of the protein environment on the fluorophore's excited-state potential energy surface and the role that the protein plays in enhancing the fluorescence quantum yield in bacterial bioluminescence.

Analytic gradients for state-averaged multiconfiguration pair-density functional theory.

Analytic gradients are important for efficient calculations of stationary points on potential energy surfaces, for interpreting spectroscopic observations, and for efficient direct dynamics simulations. For excited electronic states, as are involved in UV-Vis spectroscopy and photochemistry, analytic gradients are readily available and often affordable for calculations using a state-averaged complete active space self-consistent-field (SA-CASSCF) wave function. However, in most cases, a post-SA-CASSCF step is necessary for quantitative accuracy, and such calculations are often too expensive if carried out by perturbation theory or configuration interaction. In this work, we present the analytic gradients for multiconfiguration pair-density functional theory based on SA-CASSCF wave functions, which is a more affordable alternative. A test set of molecules has been studied with this method, and the stationary geometries and energetics are compared to values in the literature as obtained by other methods. Excited-state geometries computed with state-averaged pair-density functional theory have similar accuracy to those from complete active space perturbation theory at the second-order.

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework.

The Complete-Active Space Second-order Perturbation Theory (CASPT2) has been one of the most widely-used methods for reliably calculating electronic structures of multireference systems. Because of its lowest level treatment of dynamic correlation, it has a high computational feasibility; however, its accuracy in some cases falls short of needs. Here, as a simple yet higher-order alternative, we introduce a hybrid theory of the CASPT2 and a multireference variant of the Coupled-Electron Pair Approximation (CEPA), which is a class of high level correlation theory. A central feature of our theory (CEPT2) is to use the two underlying theories for describing different divisions of correlation components based on the full internal contraction framework. The external components, which usually give a major contribution to the dynamic correlation, are intensively described using the CEPA Ansatz, while the rests are treated at the CASPT2 level. Furthermore, to drastically reduce the computational demands, we have incorporated the pair-natural orbital (PNO) method into our multireference implementations. This development, thus, requires highly complex derivations and coding, while it has been largely facilitated with an automatic expression and code generation technique. To highlight the accuracy of the CEPT2 approach and to assess the errors caused by the PNO truncation, benchmark calculations are shown on small- to medium-size molecules, illustrating the high accuracy of the present CEPT2 model. By tightening the truncation thresholds, the PNO-CEPT2 energy converges toward the canonical counterpart and is more accurate than that of PNO-CASPT2 as long as the same truncation thresholds are used.

Analytical Gradient Theory for Quasidegenerate N-Electron Valence State Perturbation Theory (QD-NEVPT2).

Analytical nuclear gradient and derivative coupling theories for the quasidegenerate N-electron valence state perturbation theory using state-averaged orbitals and density matrices are formulated and implemented. In our implementation, the Lagrangian formalism is employed to derive the working expressions. By implementing a direct algorithm for the four-particle reduced density matrix derivatives, large active spaces up to (12e,12o) are routinely tractable. The applicability of the current algorithm is tested for optimizing the minimal energy conical intersections of the representative photochemical systems: ethylene, a retinal protonated Schiff base (penta-2,4-dieniminium cation, PSB3) model, and a green fluorescent protein chromophore (para-hydroxybenzilideneimidazolin-5-one, pHBI) model. For ethylene, we show that the optimized geometries reasonably agree with the previous geometries using the (extended) multistate second-order complete active space perturbation theory and the multireference configuration interaction with the singles and doubles method. For PSB3, we investigate the effect of the basis set selections, ranging from cc-pVDZ to cc-pV5Z, and the effect of noninvariance in describing conical intersections. For pHBI, we test two active spaces, (4e,3o) and (12e,11o), and survey the active-space dependence. We also discuss the computational cost, the parallel efficiency, and the future applicability of the current algorithm.

Analytical Gradient Theory for Strongly Contracted (SC) and Partially Contracted (PC) N-Electron Valence State Perturbation Theory (NEVPT2).

An analytical gradient theory for single-state N-electron valence state perturbation theory (NEVPT2), using both strongly contracted (SC) and partially contracted (PC) internal contraction schemes, is developed. We demonstrate the utility of the developed algorithm in the optimization of the single-state molecular geometry of acrolein, benzyne, benzene, the retinal chromophore PSB3, the GFP chromophore pHBI, and porphine, with the cc-pVTZ basis sets. The SC-NEVPT2 analytical gradients exhibit numerical instability due to the lack of invariance with respect to the rotations among the inactive orbitals. On the other hand, PC-NEVPT2 gives molecular geometries comparable to the second-order complete active space perturbation theory in any tested cases. We discuss possible future developments that will make the NEVPT2 gradient algorithm a powerful tool for optimizing molecular geometries and conducting molecular dynamics simulations of correlated systems.

Tracking Excited States in Wave Function Optimization Using Density Matrices and Variational Principles.

We present a method for finding individual excited states' energy stationary points in complete active space self-consistent field theory that is compatible with standard optimization methods and highly effective at overcoming difficulties due to root flipping and near-degeneracies. Inspired by both the maximum overlap method and recent progress in excited-state variational principles, our approach combines these ideas in order to track individual excited states throughout the orbital optimization process. In a series of tests involving root flipping, near-degeneracies, charge transfers, and double excitations, we show that this approach is more effective for state-specific optimization than either the naive selection of roots on the basis of energy ordering or a more direct generalization of the maximum overlap method. We provide evidence that this state-specific approach improves the performance of complete active space perturbation theory for vertical excitation energies. Furthermore, we find that the state-specific optimization can help avoid state-averaging-induced discontinuities on potential energy surfaces. With a simple implementation, a low cost, and compatibility with large active space methods, the approach is designed to be useful in a wide range of excited-state investigations.

Perturbation Expansion of Internally Contracted Coupled-Cluster Theory up to Third Order.

The internally contracted multireference coupled-cluster (icMRCC) method is analyzed through third order in perturbation theory. Up to second order, the icMRCC perturbation expansion is equivalent to that of the standard Rayleigh-Schrödinger perturbation theory, which is based on a linear ansatz for the wave function, and the resulting theory is, depending on the employed zeroth-order Hamiltonian, equivalent to either second-order complete active space perturbation theory (CASPT2), N-electron valence perturbation theory (NEVPT2), or Fink's retention of the excitation degree perturbation theory (REPT2). At third order, the icMRCC perturbation expansion features additional terms in comparison to the Rayleigh-Schrödinger perturbation theory, but these are shown to be nearly negligibly small by both analytic arguments and numerical examples. Considering these systematic cancellations, however, may be important in future work on approximations to icMRCC theory. In addition, we provide an extensive set of tests of the second and third-order perturbation theories based on three different zeroth-order Hamiltonians, namely, the projected effective Fock operator as used for CASPT, the Dyall Hamiltonian as used for NEVPT, and the Fink Hamiltonian used for REPT. While the third-order variant of REPT often gives absolute energies that are rather close to values from higher level calculations, the results for relative energies and spectroscopic constants such as harmonic frequencies, give a less clear picture and a general conclusion about any best zeroth-order Hamiltonian does not emerge from our data. For small active spaces, REPT is rather prone to intruder state problems.

Zero-Field Splitting Parameters from Four-Component Relativistic Methods.

We report an approach for determination of zero-field splitting parameters from four-component relativistic calculations. Our approach involves neither perturbative treatment of spin-orbit interaction nor truncation of the spin-orbit coupled states. We make use of a multi-state implementation of relativistic complete active space perturbation theory (CASPT2), partially contracted N-electron valence perturbation theory (NEVPT2), and multi-reference configuration interaction theory (MRCI), all with the fully internally contracted ansatz. A mapping is performed from the Dirac Hamiltonian to the pseudospin Hamiltonian, using correlated energies and the magnetic moment matrix elements of the reference wave functions. Direct spin-spin coupling is naturally included through the full 2-electron Breit interaction. Benchmark calculations on chalcogen diatomics and pseudotetrahedral cobalt(II) complexes show accuracy comparable to the commonly used state-interaction with spin-orbit (SI-SO) approach, while tests on a uranium(III) single-ion magnet suggest that for actinide complexes the strengths of our approach through the more robust treatment of spin-orbit effects and the avoidance of state truncation are of greater importance.

Valence ππ* Excitations in Benzene Studied by Multiconfiguration Pair-Density Functional Theory.

We explore the valence singlet and triplet ππ* excitations of benzene with complete active pace self-consistent field (CASSCF) theory, complete active space perturbation theory (CASPT2), and multiconfiguration pair-density functional theory (MC-PDFT) for four different choices of active space. We propose a new way to quantify the covalent and ionic character of the electronic states in terms of the components of the total electronic energy. We also explore the effect of scaling the exchange and correlation components of the on-top density functional used in MC-PDFT; we observe that increasing the exchange contribution improves the MC-PDFT excitation energies for benzene.

Molecular interactions in electron-groups embedding generalized valence bond picture

Theoretical description of molecular interactions remains a challenge for computational chemistry. In particular, systems dominated by static correlation, in which bonds are stretched or twisted, are often beyond capabilities of methods based on a single-electron approximation, being usually a method of choice. For interacting multireference systems, it is necessary to apply either high-level coupled-cluster methods (which however include multireference effects only partially) or base the theoretical description on a multireference wavefunction. Popular multireference methods like CASPT2 (complete active space perturbation theory) do not provide satisfactory results since they may suffer from problems with size consistency and poor accuracy. Recently we have shown that combining a simple multireference wavefunction, perfect-pairing generalized valence bond (GVB) with extended random phase approximation in embedding framework (EERPA) leads to a method EERPA-GVB providing accurate results for challenging multireference systems. In this paper, good performance of EERPA-GVB is confirmed by its application to van der Waals and hydrogen-bonded complexes. In addition, we show that the decomposition of the EERPA-GVB correlation energy into contributions from pairs of geminals can provide useful insight into the investigated interactions.

Surface hopping dynamics including intersystem crossing using the algebraic diagrammatic construction method.

We report an implementation for employing the algebraic diagrammatic construction to second order [ADC(2)] ab initio electronic structure level of theory in nonadiabatic dynamics simulations in the framework of the SHARC (surface hopping including arbitrary couplings) dynamics method. The implementation is intended to enable computationally efficient, reliable, and easy-to-use nonadiabatic dynamics simulations of intersystem crossing in organic molecules. The methodology is evaluated for the 2-thiouracil molecule. It is shown that ADC(2) yields reliable excited-state energies, wave functions, and spin-orbit coupling terms for this molecule. Dynamics simulations are compared to previously reported results using high-level multi-state complete active space perturbation theory, showing favorable agreement.

Combined complete active space configuration interaction and perturbation theory applied to conformational energy prototypes: Rotation and inversion barriers

A second-order multireference perturbation theory, termed as IVO-SSMRPT which allows the use of CASCI reference wave functions with improved virtual orbitals (IVO) for capturing static correlation and state-specific parameterization of the state-universal electronic wave function in an attempt to account for dynamic correlation has been utilized in an investigation of the torsional properties of ethylene, silaethylene, hydrogen peroxide, hydrazine, and oxalyl chloride. We also calculate the barrier to inversion of ammonia. IVO-SSMRPT is robust and useful to scan energy surfaces as it avoids the intruder-state problem, a troubling aspect of various established MRPT methods, without exploiting level-shifting or increasing the size of the active space. We find that IVO-SSMRPT with the use of a relatively small active space and basis set can be compared with recent reference estimates which are reproduced within the expected precision indicating the method is useful for the study of rotation and inversion barriers of challenging molecules.

A multiconfigurational approach to the electronic structure of trichromium extended metal atom chains

Density functional theory, Complete Active Space Self-Consistent Field (CASSCF) and perturbation theory (CASPT2) methodologies have been used to explore the electronic structure of a series of trichromium Extended Metal Atom Chains (EMACS) with different capping ligands. The study is motivated by the very different structural properties of these systems observed in X-ray experiments: the CN--capped example has a symmetric Cr3 unit while for the NO3--capped analogue the same unit has two very different Cr-Cr bond lengths. Density functional theory fails to locate an unsymmetric minimum for any of the systems studied, although the surface corresponding to the asymmetric stretch is very flat. CASPT2, in contrast, does locate an unsymmetric minimum only for the NO3--capped system, although again the surface is very flat. We use the Generalized active space (GASSCF) technique and effective Hamiltonian theory to interrogate the multi-configurational wavefunctions of these systems, and show that the increase in the σ-σ* separation as the chain becomes unsymmetric plays a defining role in the stability of the ground state and its expansion in terms of configuration state functions.

Paramagnetism of Aqueous Actinide Cations. Part II: Theoretical Aspects and New Measurements on An(IV).

The magnetic properties of actinide(IV) (An(IV)) cations are investigated in various solutions (HClO4, HCl, and HNO3) by the Evans NMR method. The magnetic susceptibilities measured in noncomplexing medium are compared with the previous studies, and the influence of the medium is verified with new measurements in complexing solutions. To rationalize these results, spin-orbit complete active space perturbation theory at second order calculations are performed on the free ions and on the aquo complexes to determine the nature of electronic states, the magnetic susceptibility, and the UV-visible-near-IR spectra. The different factors contributing to the An(IV) magnetic properties were identified. The ligand field effect on the magnetic behavior (Curie constant and temperature-independent susceptibilities) was analyzed by considering different solvation environments. These results indicate a significant effect of the zero-field splitting of the ground J manifold on the An(IV) magnetic susceptibility.

Nuclear Energy Gradients for Internally Contracted Complete Active Space Second-Order Perturbation Theory: Multistate Extensions

We report the development of the theory and computer program for analytical nuclear energy gradients for (extended) multistate complete active space perturbation theory (CASPT2) with full internal contraction. The vertical shifts are also considered in this work. This is an extension of the fully internally contracted CASPT2 nuclear gradient program recently developed for a state-specific variant by us [MacLeod and Shiozaki, J. Chem. Phys. 2015, 142, 051103]; in this extension, the so-called λ equation is solved to account for the variation of the multistate CASPT2 energies with respect to the change in the amplitudes obtained in the preceding state-specific CASPT2 calculations, and the Z vector equations are modified accordingly. The program is parallelized using the MPI3 remote memory access protocol that allows us to perform efficient one-sided communication. The optimized geometries of the ground and excited states of a copper corrole and benzophenone are presented as numerical examples. The code is publicly available under the GNU General Public License.