Derivatives response equations

The pA2 calculation is derived by equating the response produced by the full agonist in the absence of the inverse agonist (Equation 6.64 with [B] = 0) to the response in the presence of a concentration of the inverse agonist that produces a dose ratio of 2 (by definition the pA2). For calculation of KB from 10-pAT... 

The relative response of a single CSTR to an ideal pulse input may be obtained by taking the time derivative of equation 11.1.13. 

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

Both A and Ap(r) depend on the perturbation 8ucxl(r). The linear response x (Equation 24.4) is obtained by functional derivative of Equation 24.44 ... 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

To determine the response of the condensate to an external electromagnetic field we derive field equations by minimizing the free energy with respect to... 

When binding of a substrate molecule at an enzyme active site promotes substrate binding at other sites, this is called positive homotropic behavior (one of the allosteric interactions). When this co-operative phenomenon is caused by a compound other than the substrate, the behavior is designated as a positive heterotropic response. Equation (6) explains some of the profile of rate constant vs. detergent concentration. Thus, Piszkiewicz claims that micelle-catalyzed reactions can be conceived as models of allosteric enzymes. A major factor which causes the different kinetic behavior [i.e. (4) vs. (5)] will be the hydrophobic nature of substrate. If a substrate molecule does not perturb the micellar structure extensively, the classical formulation of (4) is derived. On the other hand, the allosteric kinetics of (5) will be found if a hydrophobic substrate molecule can induce micellization. 

The electron coupled interaction of nuclear magnetic moments with themselves and also with an external magnetic field is responsible for NMR spectroscopy. Since the focus of this study is calculation of NMR spectra within the non-relativistic framework, we will take a closer look at the Hamiltonian derived from equation (76) to describe NMR processes. In this regard, we retain all the terms, which depend on nuclear magnetic moments of nuclei in the molecule and the external magnetic field through its vector potential in addition to the usual non-relativistic Hamiltonian. The result is... 

First we must derive the equations for the perfectly stirred tanks. In these ideal tanks, it is assumed that the entire contents have the same composition as the outlet stream. Thus the C curve, or the response to a pulse input, can be found quite easily by a material balance. 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

The vacuum part of Equation (2.306) has been derived for all but the solvent contribution by Olsen et al. [69,72,73,76], The matrix representation of the solvent modifications to the response equations is found by expanding the last two terms. Generally, the solvent contributions have the following stmcture... 

We obtain by expanding the Lagrangian in orders of the perturbation along with the time-averaged procedure [51] the response functions as derivatives of the time-averaged CC quasienergy L(t) r. Finally, we obtain the response equations from the stationary condition. In particular, the linear response function is given by... 

To calculate the above integrals one must know Q(n> and R(n>. The derivatives of the rotation matrix Q are obtained by solving the appropriate response equations, as discussed in later sections. In some cases the explicit evaluation of Q(n) can be avoided by using the technique of Handy and Schaefer (1984). The expressions for Rin) in terms of the derivatives of the overlap matrix may be obtained by expanding R as... 

The first-order MCSCF response equations were first derived by Dalgaard and Jdrgensen (1978). For geometrical perturbations these equations were derived by Osamura et al. (1982a) using a Fock-operator approach. 

The first-order Cl response equations were derived by Jprgensen and Simons (1983) using a response function approach and reformulated by Fox et al. (1983) using a Fock-operator approach. 

Let us summarize. The calculation of Cl first anharmonicities requires no storage or transformation of second and third derivative two-electron integrals, but the full set of first derivative MO integrals is needed. One must construct and transform one set of effective density elements for third derivative integrals and 3M — 6 sets of effective densities for second derivative integrals. In addition to the 3N — 6 MCSCF orbital responses k(1) and the Handy-Schaefer vector Cm needed for the Hessian, the first anharmonicity requires the solution of 3JV — 6 response equations to obtain (1). 

Herein lies an opportunity for computing excitation spectra (and the actual CD intensity) from TDDFT linear response Once a response equation for /i(ffl) (or 4>(a>)]) has been derived, circular dichroism can be computed from an equation system that determines the poles of [> on the frequency axis, just like regular electronic absorption spectra are related to the poles of the electronic polarizability a [27]. Details are provided in Sect. 2.3. We call this the linear response route to calculating excitation spectra, in contrast to solving (approximations of) the Schrodinger equation for excited state and explicitly calculating excited state... 

For an alternative formulation of TDDFT response equations starting with a definition of response properties as quasi-energy derivatives, the reader is referred to [50] and references provided therein. 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

These equations are known as the response equations since they determine the first derivatives (i.e., the first-order response) of the wave function to the perturbation. 

Derivation of Response Equations for Quantum-Classical Systems 369... 

DERIVATION OF RESPONSE EQUATIONS FOR QUANTUM-CLASSICAL SYSTEMS... 

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