Static quantum response

Here we have used the notation for the Kubo transform of an operator X defined by 

When the external force F is small and constant we again seek a linear dependence of the form 

together with (11.33) imply that (11.32) holds. 

Problem ILL Show that in the basis of eigenstates ofHo, Ho j) = 

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Quantum linear response theory 11.2.1 Static quantum response The analogs ofEqs (11.4) are... 

Yam CY, Zheng X, Chen GH, Wang Y, Frauenheim T, Niehaus TA (2011) Time-dependent versus static quantum transport simulations beyond linear response. Phys Rev B 83 245448... 

We present here a condensed explanation and summary of the effects. A complete discussion can be found in a paper by Hellen and Axelrod(33) which directly calculates the amount of emission light gathered by a finite-aperture objective from a surface-proximal fluorophore under steady illumination. The effects referred to here are not quantum-chemical, that is, effects upon the orbitals or states of the fluorophore in the presence of any static fields associated with the surface. Rather, the effects are "classical-optical," that is, effects upon the electromagnetic field generated by a classical oscillating dipole in the presence of an interface between any media with dissimilar refractive indices. Of course, both types of effects may be present simultaneously in a given system. However, the quantum-chemical effects vary with the detailed chemistry of each system, whereas the classical-optical effects are more universal. Occasionally, a change in the emission properties of a fluorophore at a surface may be attributed to the former when in fact the latter are responsible. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

The understanding of the photoprocess responsible for the dual fluorescence can be developed in two steps. The first step is the static and structural nature and involves estimates of energy surfaces, mainly the location of minima and barriers. It is based on traditional quantum chemistry. The second step deals with dynamics of the process and requires the use of additional theoretical tools of chemical dynamics such as stochastic description. 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

The coefficients that appear in this expansion describe the response of the molecular system to the external perturbation x and are known as molecular properties. The molecular properties are characteristic of the molecular system and its quantum state. When the perturbation is static, these properties may be calculated by differentiation at x = 0... 

AODCST, 2-[4-bis(2-methoxyethyl)amino]benzylidene malononitrile PTPDac-BA2, copolymer, 65% wt N-(4-acryloyloxymethylphenyl)-N -phenyl-N,N -bis(4-methylphenyl)-[ 1,1 -biphenyl]-4,4 -diamine, 35% wt A-butylacetate DOP, dioctyl phthalate DRl-DCTA, 4,4 -di(carbazol-cl-yl)-4"-(2- N-ethyl-N-[4-(4-nitrophenyldzo)phenyl]amine ethoxy)-triphenylamine other abbreviations are defined in the text and Figures, quantum effieieney of mobile charge photogeneration has been estimated where necessary, °a relative static dielectric constant of 3 and a linear electro-optic response have been assumed. 

The molecular response tensors are characterized by peculiar properties and satisfy a series of very general quantum-mechanical relations. First we observe that the dynamic properties can be rewritten as a sum of the corresponding static property and a function multiplying the square of the angular frequency. Thus, for instance, in the case of dipole electric polarizability, using Eqn. (117),... 

The potential energy surface used in solution, G (R), is related to an effective Hamiltonian containing a solute-solvent interaction term, Vint- In the implementation of the EH-CSD model, that will be examined in Section 6, use is made of the equilibrium solute-solvent potential. There are good reasons to do so however, when the attention is shifted to a dynamical problem, we have to be careful in the definition of Vint - This operator may be formally related to a response function TZ which depends on time. For simplicity s sake, we may replace here TZ with the polarization vector P, which actually is the most important component of TZ (another important contribution is related to Gdis) For the calculation of Gei (see eq.7), we resort to a static value, while for dynamic calculations we have to use a P(t) function quantum electrodynamics offers the theoretical framework for the calculation of P as well as of TZ. The strict quantum electrodynamical approach is not practical, hence one usually resorts to simple naive models. 

The approach outlined above combines the calculation of response functions (i.e. of frequency-dependent properties) with the theory of analytic derivatives developed for static higher-order properties. In the limit of a static perturbation all equations above reduce to the usual equations for (unrelaxed) coupled cluster energy derivatives. This is an invaluable advantage for the implementation of frequency-dependent properties in quantum chemistry programs. 

The physical properties and chemical reactivity of molecules may be and often are drastically changed by a surrounding medium. In many cases specific complexes are formed between the solvent and solute molecules whereas in other cases only the non-bonded intermolecular interactions are responsible for the solvational effects. By one definition, the environmental effects can be divided into two principally different types, i.e. to the static and dynamic effects. The former are caused by the coulombic, exchange, electronic polarization and correlation interactions between two or more molecular species at fixed (close) distances and relative orientation in space. The dynamic interactions are due to the orientational relaxation and atomic polarization effects, which can be accounted for rigorously only by using time-dependent quantum theory. 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

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