MCSCF linear response

In MCSCF linear response theory [32] and the SOPPA and SOPPA(CCSD) [5,33,36] this leads to the following expression for the static dipole polarizability, e.g.,... 

The dipole and quadrupole polarizability tensor components of LiH were calculated by MCSCF linear response theory with the basis set of Roos and Sadlej [57] which consists of 13s-, 8p-, 6d-, and 2f-type sets of uncontracted Gaussian functions on Li and 12s-, 8p-, and 5d-type sets of uncontracted Gaussians on H. Due to the small size of the molecule we could perform MCSCF calculations over the whole range of internuclear distances with a very large CAS 0000 520,10,10,4 p g present the tensor components, isotropic, and anisotropic values of the dipole polarizability tensor a as function... 

However, until today no systematic comparison of methods based on MpUer-Plesset perturbation (MP) and Coupled Cluster theory, the SOPPA or multiconfigurational linear response theory has been presented. The present study is a first attempt to remedy this situation. Calculations of the rotational g factor of HF, H2O, NH3 and CH4 were carried out at the level of Hartree-Fock (SCF) and multiconfigurational Hartree-Fock (MCSCF) linear response theory, the SOPPA and SOPPA(CCSD) [40], MpUer-Plesset perturbation theory to second (MP2), third (MP3) and fourth order without the triples contributions (MP4SDQ) and finally coupled cluster singles and doubles theory. The same basis sets and geometries were employed in all calculations for a given molecule. The results obtained with the different methods are therefore for the first time direct comparable and consistent conclusions about the performance of the different methods can be made. 

In the MCSCF linear response theory [48], also called multiconfigurational RPA [54], the reference state is approximated hy a MCSCF wavefunction... 

The MPE study of the dielectric environment effects on the spin-spin coupling constants of acetylene [43] allowed for a comparison with experimentally measured gas-to-solution shifts for a series of solvents of varying polarity. It has been found in the experimental study that 1JCC changes considerably with the solvent, and that the changes correlate approximately with the solvent polarity. This tendency has been qualitatively reproduced by the MPE MCSCF linear response calculation, although the calculated changes constitute only approximately 30 % of the experimental shifts. 

The formulation of the MCSCF wavefunction in Eq. (9.47) will later be the starting point for the derivation of MCSCF linear response functions in Section 11.2. 

These are the analogous equations to the response equations for Mpller Plesset perturbation theory polarization propagators or MCSCF linear response functions in Eqs. (10.29) and (11.46). However, there are a few important differences. First, in... 

Engstrom, M., Minaev, B. R> Vahtras, O., Agren, H. (1998). MCSCF linear response calculations of electronic g-factor and spin-rotational coupling constants for diatomics. Chemical Physics Letters, 237,149. 

Fig. 12.2. The CASSCF dissociation of the C2 water molecule. On the left, we have plotted the poten-tial-eneigy curve of the ground-state valence CASSCF wave function (full line) in the cc-pVDZ basis as a function of the intemuclear OH distance. Superimposed on the ground-state curve are the potential-energy curves of the lowest excited states of the symmetry species Aj (full line), A2 (dashed line). fi (full grey line) and Bi (dashed grey line), calculated using MCSCF linear response theory. The upper plot contains the singlet excited states the lower plot contains the triplet excited states. On the right, we have plotted the lowest MCSCF Hessian eigenvalues of singlet symmetry (upper plot) and of triplet symmetry (lower plot) of the different irreducible representations. Atomic units are used.

The article is organized as follows. The main features of the linear response theory methods at different levels of correlation are presented in Section 2. Section 3 describes the calculation of the dipole and quadmpole polarizabilities of two small diatomic molecules LiH and HF. Different computational aspects are discussed for each of them. The LiH molecule permits very accurate MCSCF studies employing large basis sets and CASs. This gives us the opportunity to benchmark the results from the other linear response methods with respect to both the shape of the polarizability radial functions and their values in the vibrational ground states. The second molecule, HF, is undoubtedly one of the most studied molecules. We use it here in order to examine the dependence of the dipole and quadmpole polarizabilities on the size of the active space in the CAS and RASSCF approaches. The conclusions of this study will be important for our future studies of dipole and quadmpole polarizabilities of heavier diatomic molecules. 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

Fig. 6. HF dipole polarizability tensors (in atomic units) calculated by different linear response methods in comparison with MCSCF method.

SOPPA(CCSD) calculations with the CCSD or MCSCF PEC are also larger. In general the differences in the ZPVC are larger between the different PEC than between the different linear response methods. The SOPPA(CCSD) results for the equilibrium geometry as well as the vibrationally averaged polarizabilities are in both molecules in better agreement with the MCSCF results than the pure SOPPA values. 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

For the linear response function, we determine the modifications of the MCSCF equations due to the interactions between a quantum mechanical subsystem and a structured environment and we focus on the linear terms in ic t) and S t). For the orbital operators, c[ and we determine the modifications as ... 

The present contribution concerns an outline of the response tlieory for the multiconfigurational self-consistent field electronic structure method coupled to molecular mechanics force fields and it gives an overview of the theoretical developments presented in the work by Poulsen et al. [7, 8, 9], The multiconfigurational self-consistent field molecular mechanics (MCSCF/MM) response method has been developed to include third order molecular properties [7, 8, 9], This contribution contains a section that describes the establisment of the energy functional for the situation where a multiconfigurational self-consistent field electronic structure method is coupled to a classical molecular mechanics field. The second section provides the necessary background for forming the fundamental equations within response theory. The third and fourth sections present the linear and quadratic, respectively, response equations for the MCSCF/MM response method. The fifth 283... 

Finally, we are able to write the MCSCF/MM contributions to the linear response function as... 

Linear response theory expression Alternatively, the spin-spin coupling constant can be expressed using the linear response theory formalism. Let us write the electronic energy of the system perturbed by the nuclear magnetic dipole moments M/f in the form E = E(Mjf, A), where A are the variational parameters of the wave function. A may represent orbital rotation parameters for the SCF wave function, or orbital rotation parameters and coefficients of the configuration interaction expansion for the MCSCF... 

The use of Cl methods has been declining in recent years at the expense of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl- or MCSCF-based methods have been the preferred methods here. More recently linear response methods (Section 10.9) have been developed for coupled cluster wave functions, and which allow calculation of excited state properties. 

AIMD = ab initio molecular dynamics B-LYP = Becke-Lee-Yang-Parr CCSD = coupled cluster single double excitations CVC = core-valence correlation ECP = effective core potential DF = density functional GDA = gradient corrected density approximation MCLR = multiconfigurational linear response MP2 = M0ller-Plesset second-order (MRD)CI = multi-reference double-excitation configuration interaction RPA = random phase approximation TD-MCSCF = time-dependent multiconfigurational self-consistent field TD-SCF = time-dependent self-consistent field. 

The linear response methods offer a viable alternative to the Cl procedure [38]. A time-dependent (TD) perturbation theory (e.g. involving an oscillating electric field), combined with the SCF or MCSCF method is referred to as the TD-SCF (or random phase approximation, RPA) or the TD-MCSCF (or multiconfigurational linear response, MCLR), respectively. Let us consider the time development of the dipole moment (z-component for simplicity) ... 

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