CASSCF Wave Function Research Articles

Similarities between single reference perturbation theory based on a CASSCF wavefunction and multireference perturbation theory based on a reference space spanned by a CAS

The first-order wave-operator Ω (1) α for state specific perturbative methods, based an a single-reference state CASSCF (or CASSCI) wavefunction | α〉, is shown to be similar to the first-order wave-operator Ω (1) for multireference perturbation theory (MRPT) when the zeroth-order states {| p〉} that span the reference space are forced degenerate with energy E α 0, where E α 0 is the zeroth-order energy of the CASSCF (or CASCI) state of interest | α〉. Therefore, state-specific methods, like CASPT2, are similar to (forced) degenerate MRPT based on a reference space spanned by a complete active space (CAS).

P-Phenylenebis(methylene): a CASSCF study of σ2p2, σ3p1 and σ4 electronic states

The 1A g, 3B u and 5A g ( σ 2 p 2), 3A′′ ( σ 3 p 1), and 1A g ( σ 4) electronic states of p-phenylenebis(methylene) are studied by means of ten-electron ten-orbital CASSCF calculations. The σ 2 p 2( 1A g) state is found to be the electronic ground state and it can be described as a 1(σ,σ) diradical. The associated 3(σ,σ) triplet ( 3B u) and the quintet 55A g states are located 1.9 kcal mol −1 and 25.2 kcal mol −1, respectively, above the ground state. Both the 1A g and the 3B u electronic states exhibit a marked quinonoidal distortion. The electronic states that do not involve a local 3σp configuration on each carbene unit ( σ 3p 1 ( 3A′′) and σ 4 ( 1A g) ) are found to be higher in energy (37.0 kcal mol −1 and 59.3 kcal mol −1, respectively). The former ( 3A′′) can be described as resulting from the coupling of a σ 2 and a 3σp carbene. Finally, the CASSCF wave functions of the three nearly diradical states ( 1A g, 3B u and 3A′′) are analyzed in the light of Salem's classification, i.e. homosymmetric and heterosymmetric diradicals.

Nonorthogonal weights of modern VB wavefunctions. Implementation and applications within CASVB

The CASVB approach is a recent development in modern valence bond theory which aims to provide easily visualized, highly accurate representations of very general types of CASSCF wavefunctions. Fully variational optimization may also be performed for quite general types of modern valence bond wavefunction. Numerous definitions of nonorthogonal weights are used to investigate the role of contributions from various types of ‘ionic’ configurations, taking hexatriene and N2S2 as illustrative examples.

The X˜ 1 A 1 , a˜ 3 B 1 , a˜ 1 B˜ 1 , and B˜ 1 A 1 electronic states of SiH 2

Four electronically low-lying states of silylene (SiH2) have been studied systematically using high level ab initio electronic structure theory. Self-consistent field (SCF), two-configuration (TC) SCF, complete active space (CAS) SCF, configuration interaction with single and double excitations (CISD), and CASSCF second-order (SO) CI levels of theory were employed with eight distinct basis sets. The zeroth-order wave functions of the ground (X˜ 1A1 or 1 1A1) and B˜ 1A1 (or 2 1A1) excited states are appropriately described by the first and second eigenvectors of the TCSCF secular equations. The TCSCF-CISD, CASSCF, and CASSCF-SOCI wave functions for the B˜ 1A1 (or 2 1A1) state were obtained by following the second root of the CISD, CASSCF, and SOCI Hamiltonian matrices. At the highest level of theory, the CASSCF-SOCI method with the triple zeta plus triple polarization augmented with two sets of higher angular momentum functions and two sets of diffuse functions basis set [TZ3P(2f,2d)+2diff], the energy separation (T0) between the ground (X˜ 1A1) and first excited (a˜ 3B1) states is determined to be 20.5 kcal/mol (0.890eV,7180cm−1), which is in excellent agreement with the experimental T0 value of 21.0 kcal/mol (0.910eV,7340cm−1). With the same method the T0 value for the a˜ 1B1−X˜ 1A1 separation is predicted to be 45.1 kcal/mol (1.957 eV,15780 cm−1), which is also in fine agreement with the experimental value of 44.4 kcal/mol (1.925 eV,15530 cm−1). The T0 value for the B˜ 1A1−X˜ 1A1 separation is determined to be 79.6 kcal/mol (3.452 eV,27 840 cm−1). After comparison of theoretical and experimental T0 values for the a˜ 3B1 and a˜ 1B1 states and previous studies, error bars for the B˜ 1A1 state are estimated to be ±1.5 kcal/mol (±525 cm−1). The predicted geometry of the B˜ 1A1 state is re(SiH)=1.458 and θe=162.3∘. The physical properties including harmonic vibrational frequencies of the B˜ 1A1 state are newly determined.

Electron correlation on torsional and inversion barrier heights of acetaldehyde and acetone in the S 1(n,π ∗) state: CASSCF and CAS-MP2 study

The torsional and inversion barrier heights of acetaldehyde and acetone in the S 1 state were studied using CASSCF and CAS-MP2 wavefunctions, where the geometries of the conformers were optimized at the CASSCF level and the dynamical electron correlation energies were evaluated by the multiconfigurational second-order perturbation theory (CAS-MP2). The CASSCF wavefunction largely overestimates the inversion barrier height, while it slightly underestimates the torsional barrier height. The second-order correction to the torsional barrier height is positive, but that to the inversion barrier height is negative. The results of this study suggest that the nondynamical electron correlation, especially the correlation of π and 3p x (O) electrons for the torsional barrier and the dynamical electron correlation for the inversion barrier, is important. Calculated torsional and inversion barrier heights agreed reasonably with experimental results.

Use of symmetry-adapted Brillouin theorem to analyze the variational content of molecular wave functions along potential energy surfaces: Application to BH2 and PO2

The analysis of the Brillouin conditions in a chain of groups is used to explain why self-consistent field (SCF), multiconfigurational SCF (MCSCF), or configuration interaction (CI) methods can lead to an unbalanced description of the potential energy hypersurfaces along symmetry-broken pathways. A Wigner-Racah formulation of the generalized Brillouin theorem is the appropriate theoretical tool for performing such an analysis. We apply this procedure to a Renner-Teller splitting in BH2 and to a pseudo-Jahn-Teller conical intersection in PO2. A systematic ab initio study of both systems leads to the definition of compact CASSCF wave functions which ensure a balanced description of the ground Born-Oppenheimer potential energy surfaces. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 521–541, 1997

Fully variational optimization of modern VB wave functions using the CASVB strategy

We outline the CASVB strategy, which may be used either to generate very compact modern valence bond representations of CASSCF wave functions or to optimize general types of modern VB wave function. A simple algorithm is presented for the elimination of redundant or constrained parameters from the appropriate second-order optimization problems. Selected results for three systems—benzene, the allyl radical, and LiH—are used to illustrate various ways in which CASVB procedures can be used to perform fully variational optimizations of nonorthogonal orbitals and structure coefficients in modern VB wave functions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 439–451, 1997

Ab initio molecular orbital calculations by the resonating Hartree-Fock approach: superposition of non-orthogonal Slater determinants

The resonating Hartree-Fock (ResHF) method is applied to ab initio MO calculations of CO. This method approximates correlated wavefunctions by superposition of non-orthogonal Slater determinants, where both the CI coefficients and the orbitals in the determinants are variationally determined. The ResHF, with two or three generating determinants, explains the correlation energy to the same extent as the full-valence CASSCF wavefunction with 328 CSFs. Furthermore, the method reproduces the dipole moment more accurately, reflecting a better long-range behavior of the wavefunction. This implies the ResHF method provides more suitable reference functions in multireference methods.

Modern valence-bond description of the electronic structure of benzocyclobutadiene

The ground state of benzocyclobutadiene, a bicyclic molecule with 8π electrons containing a benzene and a cyclobutadiene ring, is studied by means of modern valence bond (VB) theory in its spin-coupled (SC) form and the complete-active-space self-consistent field (CAS SCF) approach. The CAS SCF wave function is used to optimize the geometry, and SC theory—to obtain a well-correlated and easy to visualize and understand model of the active space hosting the π electrons. It is shown that the π system of the ground state of benzocyclobutadiene can be described with sufficient accuracy as a combination of the π systems of a distorted benzene ring and an isolated double bond. Each of the eight SC orbitals is found to be well-localized about one carbon atom only, with small distortions toward its nearest neighbors. The analysis of the optimal active-space spin-coupling pattern within the SC wave function for benzocyclobutadiene shows unambiguously that this molecule inherits neither the aromatic nor the antiaromatic character of its cyclic components, and it is most appropriate to regard it as nonaromatic. © 1996 John Wiley & Sons, Inc.

Exact transformations of CI spaces, VB representations of CASSCF wavefunctions and the optimization of VB wavefunctions

It is demonstrated how exact transformations of full CI spaces may be carried out for general, non-unitary orbital transformations, and a detailed description of an efficient implementation of this scheme is presented for Slater determinants. It is then shown how this technology may be employed in the optimization of general VB wave functions with respect to both orbital and structure coefficients. This may be done in a straightforward manner, incorporating first and second derivatives of the variational parameters.

Geometry and electronic structure of model small nitrones and their tautomers

Ab initio MO studies have been performed on the parent nitrone [CH 2N(H)O], its tautomers and methyl derivatives. The effect of non-dynamical electron correlation has been taken into account using the CASSCF wavefunction. Different kinds of excitation patterns have been applied in the (2,2) to (12,12) range in order to appreciate the effect of proper orbital selection. Not only the π (a″) orbitals were found to be essential for a reliable description of nitrone molecular properties but also certain σ (a′) ones have also been shown to play an important role. Calculating the Mayer's bond indices and performing localization studies revealed four fully formed covalent bond and a fifth one which could be regarded either as a covalent bond polarised into the N → O bond direction or as a partially back-donated lone pair into the O → N direction. Comparing the calculated geometrical parameters of the parent nitrone to those of its oxime and nitroso tautomers (using several methods from HF/6–31G to CCSD(T)/TZP and MR-CISD/6–31IG ∗ levels of theory) also points to this direction. Since from CASSCF studies and natural orbital analyses the biradical contribution was found to be small, the electronic structure of the nitrone can be regarded as intermediate between the hypervalent and zwitterionic forms.

Electric properties of the water molecule in 1A1, 1B1, and 3B1 electronic states: Refined CASSCF and CASPT2 calculations

The dipole moments and dipole polarizabilities of the 1A1, 1B1, and 3B1 electronic states of the water molecule have been calculated by using the CASSCF approach followed by the evaluation of the dynamic electron correlation contribution by the second-order perturbation scheme CASPT2. All calculations have been carried out in a specifically extended ANO basis set which accounts for the Rydberg character of the two excited states. In order to estimate the correctness and accuracy of the present data a scan over a variety of different active spaces for the CASSCF wave function has been made. The present results are superior to earlier CASSCF calculations, although their qualitative features remain essentially the same. The dipole moments in 1B1 and 3B1 states are predicted to be about 0.49 a.u. and 0.33 a.u., respectively, and have the opposite orientation with respect to the ground state dipole moment. The dipole polarizability tensors of the excited states are characterized by high anisotropy and are dominated by the in-plane component perpendicular to the symmetry axis. All their components are found to be about an order of magnitude larger than those of the ground state polarizability tensor. The excitation energy dependence on the choice of the active orbital space in the CASSCF reference function is also considered and the analysis of the present data concludes in the concept of what is called the mutually compatible active spaces for the two states involved in excitation. All CASPT2 results are in good agreement with the results of recent calculations carried out in the framework of the open-shell coupled cluster formalism. This agreement confirms the high efficiency of the CASSCF/CASPT2 approach to the treatment of the electron correlation effects.

Modern valence bond representations of CASSCF wavefunctions

Exact transformations of “N electrons inN orbitals” CASSCF structure spaces are examined that lead to modern valence bond representations, in which the total wavefunction is dominated by covalent structures built from a common product ofnonorthogonal orbitals. The resulting descriptions of the electronic structure may be compared directly with those that arise in the spin-coupled (orfull-GVB) approach. Using singlet methylene, methane and ozone as representative examples, various overlap-based and energy-based criteria are investigated for generating modern VB representations of “N inN” CASSCF wavefunctions, which we denote CASVB.

Theoretical Study of the Electronic Spectrum of Imidazole

The complete active space (CAS) self-consistent field (SCF) method and multireference second-order perturbation theory (CASPT2) have been used to study the electronic spectrum of imidazole and the imidazolium ion. The calculations comprise a large number of, both singlet and triplet, valence and Rydberg excited states. A newly developed continuum model has been used to compute solvatochromic shifts. In the gas phase the first and second π → π* excited singlet valence states of imidazole are computed at 6.72 and 7.15 eV, and they shift to 6.32 and 6.53 eV upon solvation. The gas-phase values are somewhat too large (≈0.3 eV) due to an erroneous valence−Rydberg mixing in the CASSCF wave function. The first and second π → π* excited singlet valence states of the imidazolium ion are computed at 5.72 and 6.94 eV in the gas phase and shifted to 5.86 and 6.83 eV in aqueous solutions. The present results are in agreement with the observed absorption band maxima in aqueous solution, 6.0 and 6.5 eV for imidazole and 6.0 and 6.9 eV for the imidazolium ion. The computed intensities suggest another possible (but less probable) interpretation of the solution spectrum, where both species are simultaneously present. In this case the lowest band at 6.0 eV is assigned to a transition to the 21A1 state in protonated imidazole, while the second band observed in neutral solution at 6.5 eV corresponds to excitation to the 31A‘ state in imidazole.

A theoretical study of the dissociation energy of Ni 2+ A case of broken symmetry

The electronic structure and potential curves for the lowest states of the Ni 2 + cation have been studied using multiconfigurational SCF theory (CASSCF) combined with second-order perturbation theory (CASPT2) and non-orthogonal CI. The wavefunctions for the so-called σ-hole states break symmetry at all internuclear distances longer than equilibrium. CASSCF and CASPT2 calculations were first performed using C 2v symmetry. Full symmetry was restored by mixing two symmetry related CASSCF wavefunctions. Ni 2 + has a 4Σ u − ground state with a computed bond energy of 2.47 eV and a bond distance of 2.19 Å. The approach leads to a somewhat large binding energy due to double counting of the resonance energy. An estimate of the error yields a binding energy of slightly less than 2.4 eV. Calculations using D 2h symmetry give a binding energy ( D 0) of 2.25 eV and a bond distance of 2.22 Å, in apparent agreement with experiment (D 0 = 2.25 eV, r e = 2.22 A ̊ ) , indicating that the symmetry breaking error is independent of the internuclear distance.

Ab Initio Study of the Photochemical Dissociation of Methylamine

The threshold photochemistry of methylamine along three dissociation channels is examined using CASSCF and MRSDCI wave functions. Excitation of a nitrogen lone pair electron into a Rydberg s orbital produces an excited state that can decompose via NH rupture, CN rupture, or CH rupture. In the case of NH and CN rupture the excited state and ground state surfaces cross as the molecule dissociates, forming a conical intersection. The NH rupture excited state products lie above the excitation energy, and so dissociating molecules are funneled through the conical intersection and emerge as ground state products. The CN rupture excited state products lie below the excitation energy, and dissociation would proceed mostly on the excited state surface. In the case of CH rupture, the excited state products lie above the excitation energy, but since there is no conical intersection in this channel, CH rupture has to occur only after internal conversion to the ground state. These three mechanisms are all consistent w...

Exact transformations of CI spaces, VB representations of CASSCF wavefunctions and the optimization of VB wavefunctions

Exact transformations of CI spaces, VB representations of CASSCF wavefunctions and the optimization of VB wavefunctions

Modern valence bond representations of CASSCF wavefunctions

Modern valence bond representations of CASSCF wavefunctions

A CASPT2 calculation of the lowest excited states of H 2Fe(CO) 4

CASPT2 calculations based on CASSCF reference wavefunctions, using large atomic natural orbital basis sets, are reported for the lowest excited states of H 2Fe(CO) 4. The excitation energies obtained with this accurate method are compared to the values from multireference contracted CI calculations based on a unique CASSCF wavefunction. This computational strategy is usually employed for the calculation of the one-electron properties, spin-orbit interaction or costly potential energy surfaces describing the photochemistry of organometallic molecules. The relative order of the excited states is not affected by this more refined treatment and the largest excitation energy differences are of the order of 0.25 eV for the lowest excited states. However, the computational strategy seems to have a dramatic influence on the results obtained for the highest excited states pointing out the limit of less accurate methods. The theoretical spectra obtained for this family of molecules must be compared with poorly resolved experimental spectra.

Practical remarks on the selection of the active space in the CAS-SCF wavefunction

The effect of the separation of an N-electron molecular system into two subsystems, as in the CAS-SC wavefunction, is presented and analyzed from both geometrical and electronic structure points of view. The MRD-CI natural orbital selected in an adequate way can be used to build a CAS-SCF wavefunction which incorporates only the strong electron correlation. We give as an example the electronic structure of carbonyl oxide.