Wavefunction instability

To expand a bit on Dewar s cautious endorsement of the SCF procedure [20] ( SCF calculations are by no means foolproof . ..Usually one finds a reasonably rapid convergence to the required solution ) occasionally a wavefunction is obtained that is not the best one available from the chosen basis set. This phenomenon is called wavefunction instability. To see how this could happen note that the SCF method is an optimization procedure somewhat analogous to geometry optimization (Section 2.4). In geometry optimization we seek a relative minimum or a transition state on a hypersurface in a mathematical energy versus nuclear coordinates space defined by E =/(nuclear coordinates) in wavefunction... 

This indicates that there is a UHF wavefunction which is lower in energy than the RHF wavefunction, which is what we expect in this case. Note that instabilities can be of many different types. The most common kinds are ... 

The lowest energy wavefunction is a singlet, but not a dosed shell singlet (e.g., biradicals). This is an RHF-to-UHF instability. 

There is more than one solution to the SCF equations for the system, and the calculation procedure converges to a solution which is not the minimum (often a saddle point in wavefunction space). This indicates an RHF-to-RHF or UHF-to-UHF instability, depending on the wavefunction type. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

Cramer considers the appreciation of how DFT performs with diradicals as the major theoretical contribution of the work on benzynes. DFT accommodated much more multireference character in singlet wavefunctions of diradicals than did HF. But restricted DFT breaks down with p-benzyne. People were just beginning to play with UDFT and recognize the interpretation of Professor Dieter Cremer deserves more credit for this than anyone else. My contribution was recognizing that the high symmetry of p-benzyne allows for CC to work properly. But with lower symmetry systems, like pyridynium, CCSD gives wacky results but Bruckner orbitals remove some instabilities and so it works well. ... 

If these can be tested directly, even inspired empirical estimates of correlated wave-functions may become a valuable subject. At a more realistic level, however, the resolution of instabilities in predictions of excited states the assessment of the claims of different systematic calculations and the assessment of rates of convergence with different systems of expansion functions will probably form the major use of the direct assessment of the accuracy of correlated wavefunctions. 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

The performance of the RB-CCD method (which is analogous to the conventional unrestricted B-CCD method) has been tested on the nitrate radical, NO3, and the C A2 state of NO2, both of which have presented difficulties for a variety of theoretical methods due in part to symmetry-breaking instabilities in the Ffartree-Fock reference wavefunction. The RB-CCD method was found to provide results in excellent agreement with the B-CCD method, including the correct prediction of C symmetry for the equilibrium geometry of the C state of NO2 Ajyork is presently under way for extension of the RB-... 

T. D. Crawford, J. F. Stanton, W. D. Allen, and H. F. Schaefer, /. Chem. Phys., 107, 10626 (1997). Hartree-Fock Orbital Instability Envelopes in Highly Correlated Single-Reference Wavefunctions. 

Compared with the situation for the 3d series, the 4d orbitals are relatively more diffuse and hence can overlap more efficiently. One result of this diffuseness is the absence of permanent magnets amongst the 4d solids the 4d wavefunctions cannot support local moments. This can be rationalized in terms of the Stoner theory of itinerant magnetism which also has a cluster or molecular analog L Briefly, a magnetic instability may occur if... 

Those indices make it possible to discuss the instability in chemical bonds from the BS wavefunctions without the SCE. In addition, one can utilize the indices to estimate the contribution of double excitation for very large systems in which CAS and MR methods cannot be applied. In this chapter, magnetic orbitals that correspond to the CAS space for the static correlation correction are examined by the use of fe(AP). 

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