Frozen orbital method

Murphy et al. [34,45] have parameterized and extensively tested a QM/MM approach utilizing the frozen orbital method at the HF/6-31G and B3LYP/6-31G levels for amino acid side chains. They parameterized the van der Waals parameters of the QM atoms and molecular mechanical bond, angle and torsion angle parameters (Eq. 3, Hqm/mm (bonded int.)) acting across the covalent QM/MM boundary. High-level QM calculations were used as a reference in the parameterization and the molecular mechanical calculations were performed with the OPLS-AA force... 

Of the three types of boundary treatment, the link atom method is the simplest both conceptually and in practice, and is hence the most widely used. The boundary atom and in particular the frozen orbital methods can potentially achieve higher accuracy but require careful a priori parametrization and bear limitations on transferability (Senn and Thiel 2009). 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

Separation of covalently bonded atoms into QM and MM regions introduces an unsatisfied valence in the QM region this can be satisfied by several different methods. In the frozen-orbital approach a strictly localized hybrid sp2 bond orbital containing a single electron is used at the QM/MM junction [29]. Fro-... 

The first reported approach along these lines was the localized self-consistent-field (LSCF) method of Ferenczy et al. (1992), originally described for the NDDO level of theory. In this case, the auxiliary region consists of a single frozen orbital on each QM boundary atom. 

A subtle but key difference in the methodologies is that the orbital containing the two electrons in the C-X bond is frozen in the LSCF method, optimized in the context of an X-H bond in the link atom method, and optimized subject only to the constraint that atom C s contribution be a particular sp hybrid in the GHO method. In the link atom and LSCF methods, the MM partial charge on atom C interacts with some or all of the quantum system in the GHO method, it is only used to set the population in the frozen orbitals. 

Figure 13.6 Comparison of QM/MM partitioning schemes across covalent bonds. Included MM bond stretch, angle bend, and torsion tenns are indicated those that are boxed are ignored by some authors. Frozen orbitals are in gray for the LSCF and GHO methods...

The most accurate theoretical results for positronium formation in positron-helium collisions in the energy range 20-150 eV are probably those of Campbell et al. (1998a), who used the coupled-state method with the lowest three positronium states and 24 helium states, each of which was represented by an uncorrelated frozen orbital wave function... 

A reduced variational space method, related to the KM procedure, has been developed in which the orbitals of one fragment are optimized in the field of the frozen orbitals of its partner. Truncation of the variational space by deletion of unoccupied orbitals of one partner or the other is the pathway to evaluation of polarization, charge-transfer, and BSSE terms. When applied to the water dimer -", the Coulomb and exchange sum dominates the interaction but charge transfer and polarization terms are needed for proper angular dependence. 

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

Most theoretical calculations of x-ray emission rates in atoms and molecules have been performed in the frozen-orbital (FR) approximation, where the same atomic or molecular potential is used before and after the transition. It is usual to use the ground state configuration for this purpose. This approximation is convenient because we need only one atomic or molecular calculations and the wave functions for the initial and final states are orthogonal. However, the presence of vacancy is not taken into consideration. On the other hand, we have shown that the TS method is useful to predict x-ray transition energies, but is not so good approximation to the absolute x-ray transition probabilities [39]. 

While the multiconfiguration methods lead to large and accurate descriptions of atomic states, formal insight that can lead to a productive understanding of structure-related reaction problems can be obtained from first-order perturbation theory. We consider the atomic states as perturbed frozen-orbital Hartree—Fock states. It is shown in chapter 11 on electron momentum spectroscopy that the perturbation is quite small, so it is sensible to consider the first order. Here the term Hartree—Fock is used to describe the procedure for obtaining the unperturbed determi-nantal configurations pk). The orbitals may be those obtained from a Hartree—Fock calculation of the ground state. A refinement would be to use natural orbitals. 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

The local self-consistent field (LSCF) method108 provides a clear and consistent framework for treating the boundary between covalently bonded QM and MM atoms. In the LSCF method, a strictly localized bond orbital, also often described as a frozen orbital, describes the electrons of the frontier bond. This frozen orbital is used at the QM/MM boundary, i.e. for the QM atom at the frontier between QM and MM regions. The electron density of the orbital is... 

The use of frozen orbitals, such as the bond orbitals connecting the quantum to the classical part of the system, can be extended to nonempirical quantum methods such as ab initio Hartree -Fock, post Hartree Fock, or DFT. In these cases, the overlap between atomic orbitals is taken into account and the orthogonality conditions are more difficult to fulfill. The mathematical formulation of the method has been developed in the original papers [26 28] and the process can be summarized as follows. 

Murphy RB, Philipp DM, Friesner RA. Frozen orbital QM/MM methods for density functional theory. Chem Phys Lett 2000 321 113-120. 

Far from the ionisation threshold, where the escaping photoelectron has a large kinetic energy, the normal method of calculation for the continuum states is to compute the orbitals in a field determined by using the frozen orbitals of the neutral atom with one electron removed. However, if one is calculating a resonance which lies close to the threshold, this approach may fail. This happens because the escaping photoelectron moves slowly, so that the residual ion has time to relax as it escapes. It is then better to compute the orbitals in the relaxed field of the ion. This approximation is called the GRPAE or the RPAER, and is referred to as the RPAE with relaxation. 

The students are given exercises that involve the computation of ionization energies for a series of two-electron atomic systems by both the AE method (allowing orbital relaxation) and by means of Koopmans theorem (frozen orbital) they compare their theoretical results to experimental results in a written report. Koopmans theorem does well at this level, so we need to tell them about cancellation of errors This is an important lesson in science, in that good agreement with experiment does not necessarily mean good theory. ... 

The composition of this review is as follows Section 2 describes the numerical examples of the rules for degenerate excitations. The data in the next section are obtained by highly correlated methods, since the effects of electron correlations are essential for accurate descriptions of the excited states. Section 3 demonstrates the interpretation of the rules by using the simplified model that corresponds to the frozen-orbital approximation (FZOA) [4]. In the excitation energy formulas to which the FZOA leads, the splitting schemes are related to the specific two-electron integrals, whose values are qualitatively analyzed by the relevant orbital characters. Finally, the summary is addressed in Section 4. 

The IMOHC (integrated molecular orbital method with harmonic cap) method is a modification of IMOMO and IMOMM in which the bond distances to the link atom in the real system and the capping atom in the model system are not frozen but are allowed to vary in the optimization [J. C. Corchado and D. G. Truhlar, J. Phys. Chern.A,m,ms 1998)1... 

EOM-CC method (p. 638) exchange hole (p. 597) explicit correlation (p. 584) exponentially correlated function (p. 594) Fermi hole (p. 597) frozen orbitals (p. 624)... 

In the CIS method, each excited state is approximated by a linear combination of frozen-orbital single-electron excitations, First, the set of single-electron excitations is used to construct the CIS matrix, A. For spin singlet-singlet transitions. 

So far, we have mentioned methods that produce all-electron diabatic wavefunctions and corresponding Hamiltonian matrix elements. There are two other classes of methods which simplify the quantum problem by focusing on the wavefunction of the transferred charge such as methods making use of the frozen core approximation Fragment Orbital methods (FO), and methods that assume the charge to be localized on single atomic orbitals [50]. In this work, we will also treat these computationally low-cost methods. 

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