Single Perturbation Methods

The MoIIer-PIesset perturbation method (MPPT) uses the single-eonfiguration SCF proeess (usually the UHF implementation) to first determine a set of LCAO-MO eoeffieients and, henee, a set of orbitals that obey F( )i = 8i (jii. Then, using an unperturbed Hamiltonian equal to the sum of these Foek operators for eaeh of the N eleetrons =... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

If this equation is used, A A and A IS,, can be calculated from a single simulation, or, as is usually the case, from a single series of stratified simulations. This is why it is sometimes called a single-state perturbation method. Unfortunately, the problem... 

Single-step perturbation methods have also been applied to electrostatic processes. One study probed the dielectric properties of several proteins at a microscopic level [41,42], Test charges were inserted at many different positions within or around each protein, and a dielectric relaxation free energy was computed, which is related to a microscopic dielectric susceptibility (see Sect. 12.3). 

The tautomerism of furoxan (l,2,5-oxadiazole-2-oxide) has been investigated by different computational methods comprising modern density functions as well as single-reference and multi-reference ab initio methods. The ring-opening process to 1,2-dinitrosoethylene is the most critical step of the reaction and cannot be treated reliably by low-level computations (Scheme 2). The existence of cis-cis-trans- 1,2-dinitrosoethylene as a stable intermediate is advocated by perturbational methods, but high-level coupled-cluster calculations identify this as an artifact <2001JA7326>. 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

The only satisfactory approach to this problem at present involves the use of perturbations methods. Attempts to calculate resonance energies indirectly, by comparing calculated total energies of a compound with that of a single classical structure for it, have proved ineffective. Thus calculations of this kind have failed to account for the special stability of monocyclic compounds containing 4n + 2 electrons (Hiickel s rule), and they predict that all the compounds XVII-XXIII should be aromatic whereas the only one that is aromatic is azulene (XXII). 

The latter was developed as a single-configuration method, complemented by non-orthogonal configuration-interaction calculations exploiting virtual orbitals in the Spin-Coupled Valence Bond (SCVB) approach [5] (for a review, see e.g. Ref. [6]). More recently, the use of perturbative virtuals has been introduced, giving rise to the SCVB variant of the method [7] [8]. There are further accounts of SCVB in this volume. 

As a demonstration of the use of the perturbation method we refer to the work of Fogden, Mitchell and Ninham [33] who explicitly consider a single undulating charged monolayer in contact with an electrolyte. The surface is presumed to possess a sinusoidal undulation about the planar shape, but is assumed to maintain a state of constant charge ... 

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