Linear response function energy

Linear response function approaches were introduced into the chemistry literature about thirty years ago Ref. [1,2]. At that time they were referred to as Green functions or propagator approaches. Soon after the introduction it became apparent that they offered a viable and attractive alternative to the state specific approaches for obtaining molecular properties as excitation energies, transition moments and second order molecular properties. 

Koch H, Jensen HJA, Jorgensen P, Helgaker T (1990) Excitation-energies from the coupled cluster singles and doubles linear response function (CCSDLR) - applications to be, CH+, CO, and H2O. J Chem Phys 93 3345... 

They reduce to regular energy derivatives in the static limit [48,50]. The linear response function... 

A different analysis applies to the LR approach (in either Tamm-Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. 

Using the spectral representation, it is possible to obtain an informative expression for the linear response function where we denote the energy difference between the state given by n> and the reference state 0 > as con = En — E0. 

The linear response function in Eq. (11) has the same structure as the second-order energy expression in Eq. (2) and we note that for A = V and wj = 0 they are identical, except for a factor of two. Similarly, Eq. (10) defines the quadratic response function... 

We would like to stress the similarity between Eqs. (5) and (18). The main difference is that the poles of the linear response function are excitation energies rather than energy eigenvalues (c./. Eq. (11)) but in both cases, the residues correspond to transition moments between the ground state and excited states. The two-step procedure for evaluating the linear response function is now (c.f. Eqs. (6) and (7) ) solve... 

A large number of spin-orbit properties can now be derived from the response functions. From the linear response function we can deduce the second-order energy correction due to SOC (see section 4.1),... 

The change in potential energy AV is first order in the nuclear displacements, and of course, Ap is obtained correctly to the same order from equation (122). Handler and March show that the Thomas-Fermi approximation to the linear response function F has the form... 

Next we will utilize the sum over states or spectral approach when we assume that it is possible to determine an exact reference state 0) together with eigenfunctions, n>, that are solutions to a Hamiltonian given by Ho + Wse- Using this basis of exact states and that the energy difference between the state given by n> and the reference state 0> is given asco = E - Eq, we are able to write the linear response function as... 

H. Koch, H. J. Aa. Jensen, P. Jorgensen, and T. Helgaker, /. Chem. Phys., 93, 3345 (1990). Excitation Energies from the Coupled Cluster Singles and Doubles Linear Response Function (CCSDLR). Applications to Be, CH, CO, and H2O. 

Thus, the linear response function has poles where the absolute value of the frequency is equal to an excitation energy of tire system. From the corresponding residues... 

The poles of the quadratic response function are the same as those of the linear response functions, i.e. the excitation energies of the system. This is also the case for the cubic response function which, furthermore, has the same kind of residues as the quadratic response function. 

Having analysed the form of the propagators and the information obtainable from the propagators, we now wish to determine the equations of motion. They are most conveniently obtained in the time representation. The time-dependent retarded linear response function is defined in Eq. (14) and was obtained from the energy-dependent propagator (Eq. (13)) through the transformation... 

However, usually EELS can be easily measured over an extended energy range Ifom 1 eV upwards and very accurately. Together with suitable extrapolation down to 0 eV (usually with the aid of optical data), the function Im[— ] can be deduced over the whole energy range where it has significant intensity. The causality relation governing linear response functions can be used to relate Re[— ] with Im[— ] ... 

Equations (5), (11), and (12) do much to explain this reaction. For instance, consideration of the chemical potentials p and p and the hardnesses rj and rj not only allows one to predict whether or not reaction Eq. (13) will occur, but if the reaction occurs it allows one to estimate the binding energy of the product molecule, A-B [3, 12, 15, 17]. Consideration of the linear response function for the reactant molecules allows one to approximate the change in the density associated with reaction Eq. (13). However, because p and rj are global constants they contain no information on site reactivity. It is precisely information about site reactivity that is contained in the Fukui function. 

The response kernel provides a useful constraint on kinetic energy functionals because the second derivative of the noninteracting kinetic energy is related to the inverse of the linear response function... 

Both transition energies and oscillator strengths are needed for determination of optically allowed absorption spectra. In the multi-configuration version of the linear response theory (MCLR) one constructs an approximation to the exact linear response function by exposing the optimized (MC) SCF wavefunction 0> to a time-dependent perturbation. In this case the time-dependent wave function assumes the form... 

These considerations are readily extended to obtain the interaction when two disjoint distributed charges p r) andP2(f) interact. For linear response the energy (AlO) is merely summed over all 6-function pairs to find... 

Excitation Energies from the Coupled Cluster Singles and Doubles Linear Response Function (CCSDLR). Applications to Be, CH", CO, and H2O. 

This quantity can be viewed as a generalization of Fukui s frontier MO concept and plays a key role in linking Frontier MO Theory and the HSAB principle. It can be interpreted either as the sensitivity of a system s chemical potential to an external perturbation at a particular point r, or as the change of the electron density p(r) at each point r when the total number of electrons is changed. The former definition has recently been implemented to evaluate this function,but the derivative of the density with respect to the number of electrons remains by far the most widely-used definition. The second order derivative of the energy with respect to the external potential is the linear response function xCi tO called the polarizability kernel... 

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