Unrelaxed orbital

Figure 7. JB and MEB of neon and argon. Analytical response result, all electron, unrelaxed orbitals...

Any ocan be obtained from 4>o via the action of exp(Ti), provided oand are not orthogonal. This is known as the Thouless theorem.] Because the relaxed and unrelaxed orbitals are related by such an orbital rotation, we have (for CCSD),... 

In the simplest unrelaxed orbital independent particle model, the energy of this particular shake-up ionization 4 (Figure 14) would be approximated by... 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

The rotational level distributions from reactions (106) and (108) have also been determined in the low-pressure experiments [264] and those for HCl(t> = 2) are shown in Figure 1.12. The (unrelaxed) high J distribution corresponds to that produced by reaction and is clearly non-Boltzmann. Hydrides are particularly well suited to absorbing large amounts of energy as rotation because of their low moments of inertia. However, the orbital angular momentum associated with H + Cl2 collisions is unusually low, because the collisional reduced mass is small, and this apparently causes the lower cutoff in the rotational distribution from this reaction. For Cl + HI,... 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

Note that if a frozen core approximation is used a small difference will persist in the Full CI limit since the unrelaxed series converges to a limit with the core orbitals constrained to be those of the unperturbed system whereas in the relaxed series the relaxation contribution of the core orbitals is included. For electric (hyper)polarizabihties this effect is usually negligible. 

However, the equivalence of the response functions to the property derivatives is in approximate methods not always strict, as, for example, CC response functions as defined in Section 2 do not involve contributions due to orbital relaxation while property derivatives usually do. The incorporation of orbital relaxation effects in the property derivatives is mandatory when perturbation-dependent basis functions such as GIAOs/LAOs are used. Applying the above reformulation to the expressions for a(-u), w) and obtained from the CC response functions takes only relaxation with respect to the (static) external magnetic field into account [70, 71]. The frequency-dependent electric fields are treated in an unrelaxed manner, which avoids spurious poles due to orbital relaxation (see Section 2.2). 

Table 11. CCSD results for the total Verdet constant at w = 0.11391 a.u. in the case of hydrogen fluoride. Results labeled as Unrelaxed refer to the use of the unrelaxed (one-electron) magnetic dipole moment operator together with the usual magnetic-field independent basis sets. Results labeled Relaxed include additional contributions due to orbital relaxation in the presence of the magnetic field. Results labeled LAO are those obtained when using GIAOs/LAOs ...

Table 17 summarizes tire results of a benchmark study for the convergence of Atj for neon with the correlation treatment in a series of CC methods from [39], Whereas CCSf) yields results still ca. 3% too low (orbital-relaxed) or 1.4% too high (orbital-unrelaxed), the CCSDT results differs from the FCI limit by only 0.002 a.u. or a0.1%. Thus CCSDT provides for the Cotton-Mouton effect of Ne results which aie converged within approximately 0.1%. 

Figure 4. Static hypermagnetizability anisotropy. Atj(O), computed with the d-aug-cc-pV5Z basis set (Neon) and d-aug-cc-pVQZ basis set (Argon). Orbital-relaxed results obtained with a finite field approach from analytically evaluated magnetizabilities are compared to those obtained from orbital-unrelaxed quadratic and cubic response functions...

Fig. 5. EOM and Cl vertical ionization potentials for BH solid line, relaxed Cl long and short dashes, unrelaxed Cl, using SCF orbitals of BH dashed curve, extensive EOM dotted curve, primitive repartitioned EOM. The EOM results are plotted against the tolerance for retaining shake-up-basis operators in the primary operator space, and the dimension of the primary operator space is given in parentheses for each tolerance. The Cl values are presented at the one configuration level (1C), for single and double excitations Cl (SD), and for single, double, and triple excitations Cl (SDT). EOM calculations are not performed at tolerance of 0.01 au because this tolerance does not result in an appreciable increase in the dimensionality of the f -space. Experimental value is 9.77 eV. Asterisk EOM primary operator space restricted to simple ionization operators.

Two types of Cl calculation are presented. The relaxed Cl calculations employ the neutral SCF orbitals in the calculation of the ground-state energy of the neutral, and the ion SCF orbitals are used for the ion ground-state energy. The unrelaxed Cl calculations use a common set of orbitals for both the neutral and ion calculations. The orbitals are taken from a ground-state SCF calculation for whichever of the two is the closed-shell system. The unrelaxed Cl calculations are similar in this respect to the EOM calculations, which utilize only one set of closed-shell SCF orbitals throughout. Comparison of the relaxed and unrelaxed Cl calculations also affords a check of the Cl convergence. A number of different relaxed and unrelaxed Cl calculations are made for each system. The simplest involves... 

Fig. 6. EOM and Cl vertical ionization potentials for HF solid curve, relaxed Cl long and short dashes, unrelaxed Cl, using SCF orbitals of HF dashed curve, extensive EOM dotted curve, primitive repartitioned EOM. Meaning of the x-axis is the same as in Fig. 5. Experimental value is 16.01 eV. Asterisk EOM primary operator space restricted to simple ionization operators.

It is interesting to note that the Koopmans s theorem prediction for BH (i.e., the 1C, unrelaxed Cl result) is also very accurate for this basis set. This indicates that the effects of orbital relaxation and the changes in correlation approximately cancel for this example. This cancellation may be at least partially responsible for the excellent accuracy of the EOM ionization potential for BH for all systematic approximation schemes. 

FIGURE 1.9 Linear combination of atomic orbitals (LCAO) pattern of (a) the HOMO level in the neutral 6-ring OPV and the SOMO level of (b) unrelaxed and (c) fully relaxed 6-ring OPV-I-, as calculated at the INDO level on the basis of AMI-optimized geometries. The size and color of the circles reflect the amplitude and sign of the LCAO... 

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