Self-consistent field linear response

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

R. Cammi, L. Frediani, B. Mennucci, K. Ruud, Multiconfigurational self-consistent field linear response for the polarizable continuum model Theory and application to ground and excited-state polarizabilities of para-nitroanUine in solution. J. Chem. Phys. 119, 5818 (2003)... 

Self-Consistent Field Linear Response Theory and Application to Ceo. Excitation Energies, Oscillator Strengths and Frequency-Dependent Polarizabilities. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

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... 

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... 

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... 

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. 

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. 

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. 

Here, w,y((w) = tpi w o)) ipj) is the matrix element for the extra field which would have been the only contribution if the electrons were non-interacting. But since the electrons interact through Coulomb, exchange and correlation effects, an extra term describing the linear response of the self-consistent field to the perturbation has to be included. This term contains fire quantities... 

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. 

To compute the response 5, we treat each chain as ideal but subjecfed ip a potential which contains two parts. One is the external potential fFd ), the other is an internal, self-consistent, potential, due to the surrounding chains, and includes terms linear in 8. In many cases there is a strong tendency toward cancellation between the two parts this we call this screening. Finally we arrive at a self-consistent pediction for S, which is ratho accurate for concentrated chains. Thus RPA is a self-consistent field calculation for pair correlations. 

The system is now perturbed by an externally applied time-dependent (electric) field Vappi(r,t). This perturbation leads to an additional component of 5vscF(r>0, which is the linear response of the self-consistent field to the external field. The self-consistent field VscF(t>0 are the last two terms in eqn (11), i.e., the Coulomb and the exchange-correlation terms. The effective perturbation in the linear-response regime is therefore ... 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

In order to appreeiate the general eoneepts that are involved, the linear response equations for a Self-Consistent Field (SCF) ground state will be sketehed below. This description is appropriate if the state of interest is well described by a HF (Hartree-Fock) or DFT single determinant ( 2.1). The ground state energy is... 

After we obtained the self-consistent electronic structure of the magnetic multilayers we calculated the non-local conductivity by evaluating the quantum mechanical linear response of the current to the electric field using an approach developed by Kubo and Greenwood. In this approach the conductivity is obtained from a configurational average of two one-electron Green functions ... 

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