Multiconfigurational Linear Response Functions

Historically, CHF, was favoured over RPA since it could be solved in the atomic orbital basis (Diercksen and McWeeny, 1966), rather than requiring a transformation to the molecular orbital basis. The need for an inverse Hessian in RPA/SOPPA also restricted the size of system that could be studied. However, the use of direct atomic-orbital-driven methods for RPA response properties (Feyereisen et al, 1992) and for SOPPA (Bak et al., 2000 Christiansen et al., 1998a), coupled with iterative methods for solving the inverse Hessian, mean that they can now be applied as widely as CHF/TDHF and provide far more information about excited states and properties. 

In order to keep the equations more compact it is advantageous to collect the operators in one row vector = ( J),, pq, eJio On ) and correspondingly the time-dependent orbital rotation and state-transfer parameters in one column vector j t) = ( Kpg(f), pq(t), S no(i), The latter are then expanded in 

Contrary to response theory for exact states, in Section 3.11, or for coupled cluster wavefunctions, in Section 11.4, in MCSCF response theory the time dependence of the wavefunction is not determined directly from the time-dependent Schrodinger equation in the presence of the perturbation H t), Eq. (3.74). Instead, one applies the Ehrenfest theorem, Eq. (3.58), to the operators, which determine the time dependence of the MCSCF wavefunction, i.e. the operators hj  

Inserting the expression for the time-dependent MCSCF wave function, Eq. (11.36), and the perturbation expansion of the wavefunction parameters, Eq. (11.39), and separating the orders one finds for the first-order equation [see Exercise 11.5] 

Exercise 11.5 Derive the first-order equation Eq. (11.41) from the Ehrenfest theorem 

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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... 

A good overview of time-dependent response function theory, including linear and non-linear response functions is offered in J. Olsen and P. Jprgensen, Time-dependent response theory with applications to self-consistent field and multiconfigurational self-consistent field wave functions, in D. Yarkony (Ed.), Modern electronic structure theory. World Scientific Publishing, Singapore, 1995, pp. 857-990. 

Ab initio determinations of SOC in carbene (1) have been carried out for three bond angles (0 = 90, 112 and 135°) by McKellar et al. [27J using SCI and SDCI wave functions and by Vahtras et al. [19] for the equilibrium geometry employing a multiconfiguration linear response (MCLR) approach. Both sets of calculations are based on the full coupling operator with the... 

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. 

Calculation of rotational and vibrational g factors by linear response methods using multiconfigurational self-consistent-field wave functions is described in detail elsewhere [18,27]. 

R. Cammi and J. Tomasi, Nonequilibrium solvation theory for the polarizable continuum model - a new formulation at the SCF level with application to the case of the frequency-dependent linear electric-response function, Int. J. Quantum Chem., (1995) 465-74 B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798-807 R. Cammi, L. Frediani, B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, A second-order, quadratically... 

At first sight these equations do not appear to be of any use since the simple response function is merely expressed in terms of a more complicated response function of the same kind involving [P, HqI However, it is possible to obtain a closed-form expression response function, as we shall see in the linear case. At the moment little is done to reformulate Eq. (51). Olsen and Jfirgensen (1985) have shown how the quadratic and the cubic response function can be evaluated using a multiconfigurational self-consistent field (MCSCF) reference state. 

P. Jprgensen, H. Jensen, J. Olsen, Linear response calculations for large scale multiconfiguration self-consistent field wave functions, J. Chem. Phys. 89 (1988) 3654-3661. 

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... 

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