Perturbation order

The adequate technique here is to substitute the usual Rayleigh-Schrbdinger scalar perturbation order by a vector perturbation order n. 

This chapter aims to present the fundamental formal and exact relations between polarizabilities and other DFT descriptors and is organized as follows. For pedagogical reasons, we present first the polarizability responses for simple models in Section 24.2. In particular, we introduce a new concept the dipole atomic hardnesses (Equation 24.20). The relationship between polarizability and chemical reactivity is described in Section 24.3. In this section, we clarify the relationship between the different Fukui functions and the polarizabilities, we introduce new concepts as, for instance, the polarization Fukui function, and the interacting Fukui function and their corresponding hardnesses. The formulation of the local softness for a fragment in a molecule and its relation to polarization is also reviewed in detail. Generalization of the polarizability and chemical responses to an arbitrary perturbation order is summarized in Section 24.4. 

The concept of dipole hardness permit to explore the relation between polarizability and reactivity from first principles. The physical idea is that an atom is more reactive if it is less stable relative to a perturbation (here the external electric field). The atomic stability is measured by the amount of energy we need to induce a dipole. For very small dipoles, this energy is quadratic (first term in Equation 24.19). There is no linear term in Equation 24.19 because the energy is minimum relative to the dipole in the ground state (variational principle). The curvature hi of E(p) is a first measure of the stability and is equal exactly to the inverse of the polarizability. Within the quadratic approximation of E(p), one deduces that a low polarizable atom is expected to be more stable or less reactive as it does in practice. But if the dipole is larger, it might be useful to consider the next perturbation order ... 

The present formulation does not involve any global change in the number of electrons of a molecule and can be properly defined for an isolated system. We consider a variation Sp(r) induced by a potential (which does not need to be small) <5vext(r) and generalize the formula Equation 24.44 to an arbitrary perturbation order... 

The origin of the LR-SS difference was imputed to the incapability of the nonlinear effective solute Hamiltonian used in these solvation models to correctly describe energy expectation values of mixed solute states, i.e., states that are not stationary. Since in a perturbation approach such as the LR treatment the perturbed state can be seen as a linear combination of zeroth-order states, the inability of the effective Hamiltonian approach to treat mixed states causes an incorrect redistribution of the solvent terms among the various perturbation orders [32],... 

Its nontrivial solution is provided by the identity (Po)0 = 1. The corresponding term may be passed to the right-hand side of Eq. (4.264), thus rendering this equation nonhomogeneous. In all the higher perturbation orders, b(k> vanish identically. 

One possible way of limiting the complexity of such calculations regardless of the size of the system, is through the use of perturbation theory, where the maximum number of correlated electrons and the maximum number of nuclear centers that appear in the calculation are determined by the perturbation order. If such a calculation yields the desired accuracy when carried to a particular order, that order determines the maximum complexity of the calculation. One example of such a perturbation theory[69], a variant of the Z x expansion method, has been applied to Hj with promising results for such a low-order calculation. 

As the perturbation is applied onto the molecular system we represent the k(t) parameters through the perturbation orders... 

This condition means Anderson noise of large intensity, and, as we have seen, W is a weak perturbation. Note that on the extreme left and extreme right of the second term of Eq. (41) we have IIL... = — iII[W,...] and (1 — II)L... = — /(I — II) [W,...]. This means that the second term of Eq. (39) is of second order. We aim at illustrating the consequence of making a second-order approximation. To keep our treatment at the second perturbation order, we neglect the perturbation appearing in the exponential of Eq. (41). This makes the calculation of the memory kernel very easy. Using the Cauchy distribution of Eq. (33), we obtain... 

The superscripts in parentheses indicate the perturbation order, and p, q, and r label spinorbitals. We then apply an oscillatory perturbation, which can be described as a single Fourier component... 

We determine the contributions to the linear, quadratic and cubic response equations by expanding the time-dependent wave function, 0 ), and the time-dependent operator, to the appropriate perturbation order. Finally, we collect the terms related to a given order of the perturbation and we have that the terms relevant for... 

It will become clear later that at the lowest significant perturbation order, we can replace Eq. (2.15) with... 

The minimum perturbation order of each term of Eq. (3.34) can be evaluated by summing indices and exponents factor by factor. By substituting the explicit expression for I> we finally get the perturbation expansion of Eq. (3.24) at any required order in and in r. In Sections IV and V, we shall do this explicitly for some examples of interest. 

It is not necessary to invoke a complete molecular isotropy if V depends only on 0. When dV/dS = fiE sin 3, we obtain the well-known Debye equation. It is very interesting to remark that at this perturbation order there is no sign of whether the rotator is a symmetric or a spherical top. 

Ensog method applied by Titulaer, and the approach described in this chapter provide the same result up to the perturbation orders so far considered. The only reason for controversy seems to be the division of the operator into perturbed and perturbation parts. However, any choice of a perturbation part which cannot be given the form... 

This result is obtained by using corrections up to the second perturbational order for a detailed discussion of how the perturbation parameter is defined, see Chapter 1.)... 

The formal perturbation order 2n +1 is then decrea sd by at most 2/i these are precisely the terms 1/y which are to be resummed. 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

Each of these expressions is constructed by simply assigning the appropriate perturbational orders to each operator in Eq. [122] and retaining only the terms that correspond to the desired order, n. Using H as an approximate Hamiltonian, one may construct th-order Schrodinger equations of the form... 

Again the terms to be included were based on a perturbation order argument.6,16 The contribution of connected triples was shown numerically to be not inconsequential in applications of perturbation theory.19,20 More recently, Purvis and Bartlett16 reported the equations and initial implementation of a full coupled-cluster singles and doubles model (CCSD) this theory includes all terms in the first five parentheses, C0-C4, of Eq. (6) except for Ti, r4, and T Ti. The inclusion of disconnected terms is known to enhance the numerical stability of the coupled equations.21,22... 

In the next step we write Af /fy as (A// ) + (A///o) insert this expression into Eq. 97, Hnearize in (A///o), and solve for (A/Superscripts in square brackets denote the perturbation order. This step is iterated until the desired accuracy is reached. This procedure cures, for instance, the problem outhned below Eq. 95. After going through the perturbation analysis to third order, one finds ... 

The POST-C7 (permutationally offset stabilized C7) sequence applies the cycle (7r/2), 27r +,r(37r/2), which effectively cancels isotropic resonance offsets up to fourth order and cross terms between isotropic resonance offsets and RF inhomogeneity up to third order. POST-C7 thus represents an improvement by two perturbation orders compared to the original 2w 2n +, r cycle regarding both resonance-offset and cross-term compensation. The effective dipolar... 

These theoretical calculations are subject to systematic errors arising from basis set truncation and neglect of higher perturbation order electron correlation effects. Therefore, we have added a second step to our procedure in which we attempt to account for these deficiencies by adding empirically derived Bond-Additivity-Correction (BAC) factors that are based on the types of bonds present in the given molecule. The correction factors for NH (9.4 kcal-mole" ) and OH (10.7 kcal-mole" ) bonds are obtained as the difference between the theoretical and... 

Table 10.2 shows the perturbation operators arising from inclusion of a magnetic field and relativistic effects. The last three columns indicate the perturbation order with respect to an external (Bext) and internal nuclear (I) magnetic field, and with respect to the inverse speed of light. Only the most important operators up to order c have been included, with the exception of the diamagnetic nuclear spin-spin coupling operator, since the leading term for this quantity is of order c. ... 

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