Computing Potential Energy Surfaces

PES s are typically only computed for systems with a fairly small number of atoms. 

The level of theory necessary for computing PES s depends on how those results are to be used. Molecular mechanics calculations are often used for examining possible conformers of a molecule. Semiempiricial calculations can give a qualitative picture of a reaction surface. Ah initio methods must often be used for quantitatively correct reaction surfaces. Note that size consistent methods must be used for the most accurate results. The specific recommendations given in Chapter 18 are equally applicable to PES calculations. 

Once a PES has been computed, it is often fitted to an analytic function. This is done because there are many ways to analyze analytic functions that require much less computation time than working directly with ah initio calculations. For example, the reaction can be modeled as a molecular dynamics simulation showing the vibrational motion and reaction trajectories as described in Chapter 19. Another technique is to fit ah initio results to a semiempirical model designed for the purpose of describing PES s. 

Of course, the analytic surface must be fairly close to the shape of the true potential in order to obtain physically relevant results. The criteria on fitting PES results to analytic equations have been broken down into a list of 10 specific items, all of which have been discussed by a number of authors. Below is the list as given by Schatz  

The analytic function should accurately characterize the asymptotic reactant and product molecules. 

---

Static properties of some molecules ([193,277-280]). More recently, pairs of ci s have been studied [281,282] in greater detail. These studies arose originally in connection with a ci between the l A and 2 A states found earlier in computed potential energy surfaces for C2H in symmetry [278]. Similar ci s appear between the potential surfaces of the two lowest excited states A2 and B2 iit H2S or of 82 and A in Al—H2 within C2v symmetry [283]. A further, closely spaced pair of ci s has also been found between the 3 A and 4 A states of the molecule C2H. Here the separation between the twins varies with the assumed C—C separation, and they can be brought into coincidence at some separation [282]. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

Interest at Brookhaven was stirred by the contrast between the excellent agreement between theory and experiment for the D2+ + D2 reaction and the rather poor description provided for the H2-He and H2-Ne systems. The H2-He system is particularly interesting because of the relatively few particles involved in the reaction and its potential for accurate theoretical treatment. The reactions of H2 + or HD+ with He will be among the first to be treated in terms of a theoretically computed potential energy surface comparison of experiment and theory in this system is therefore of prime importance. 

Figure 4-2. Computed potential energy surface from (A) ab initio valence-bond self-consistent field (VB-SCF) and (B) the effective Hamiltonian molecular-orbital and valence-bond (EH-MOVB) methods for the S 2 reaction between HS- and CH3CI...

An alternative approximation scheme, also proposed by Born and Oppenheimer [5-7], employed the straightforward perturbation method. To tell the difference between these two different BO approximation, we call the latter the crude BOA (CBOA). A main purpose of this chapter is to study the original BO approximation, which is often referred to as the crude BO approximation and to develop this approximation into a practical method for computing potential energy surfaces of molecules. 

Ab initio calculations and density functional theory studies of the gas-phase addition of HF to CH2=CH2 have revealed the possibility of forming trimolecular (two HF and one ethylene) and dimolecular (one FIF and one ethylene) complexes and transition-state structures and of the catalytic effect of the second molecule of the reagent. An energetically favourable pathway was selected on the basis of the computed potential-energy surface for these two reactions. ... 

It is important to note that the computed potential energy surface reveals a facile reactivation via Scheme 7, with low energy barriers both in the gas phase and in aqueous solution" . This accords with experimental findings on oximates as reactivators following poisoning by organophosphorus toxics. 

A valuable approach toward the determination of solution structures is to combine molecular mechanics calculations with solution experimental data that can be related to the output parameters of force field calculations 26. Examples of the combination of molecular mechanics calculations with spectroscopy will be discussed in Chapter 9. Here, we present two examples showing how experimentally determined isomer distributions may be used in combination with molecular mechanics calculations to determine structures of transition metal complexes in solution. The basis of this approach is that the quality of isomer ratios, computed as outlined above, is dependent on the force field and is thus linked to the quality of the computed structures. That is, it is assumed that both coordinates on a computed potential energy surface, the... 

Figure 7.10. Computed potential energy surfaces of the ground state and the (n,A ) excited states T, and S, for the cis-trans isomerization of diimide as a function of the twist angle 0 and the valence angle a at one of the nitrogen atoms.

Several sources provided the data used to calculate this potential-energy surface [30, 32, 31], Moore and Pearson provide a more detailed discussion of computing potential-energy surfaces [26],... 

F re 14. Computed potential energy surfaces of isomerization of stilbene radical cations [147]. 

Wasileski et al. have validated Eqs.18 and 19 numerically by DFT-GGA cluster calculations of simple atomic adsorbates on a 13-atom Pt(l 11) cluster. The four parameters of interest, Afp) Gp, Pd(F), and -2o5d(T), were determined from DPT for four model polar adsorbates, O, Cl, I, and Na, and one model non-polar adsorbate, H, at three different fields. This allows the calculation of dKp /dF as prescribed by Eq. (18), and these values were compared with the corresponding quantities extracted directly from the field-dependent DFT-computed potential energy surfaces themselves, labeled as dK],- /dF (DF. In general, a good approximate correspondence (chiefly to within 10%) was observed between the values for dKf/dF obtained from Eq. (18) and the dKf /dF (DFT) values. This also leads to a similar correspondence between dvff /dF as obtained from Eq. (19) and dvy, /dF (DFT). The latter quantity differs only marginally from the real dv -A/dF values, which include the full anharmonicity of potential energy surface. 

Kuznetsov and Lorenz were the first to apply ab initio quantum-chemical methods to study one of the most elementary reactions in electrochemistry, the hydrogen evolution reaction. Using SCF-HF calculations they computed potential energy surfaces for the approach of a hydrogen from a solvated hydronium (i. e. H30(H20)2) to... 

The computed potential energy surfaces are often expanded in polynomials of nth (n > 3) degree and the minimum R, the zero-point frequency co. the anharmonicity co x (and corresponding terms in the other dimensions) and... 

Figure 19 shows the computed potential energy surfaces of eight electronic states of La... 

Das and Balasubramanian (199 Id) have obtained the potential-energy surfaces of 12 electronic states of HfHj. Related YH3 and ZrHj have been studied by Das and Balasubramanian (1989). They employed CASSCF followed by full second-order Cl (SOCI) using RECPs which retained the outer (5s 5p 5d 6s ) shells of Hf in the valence space. The computed potential-energy surfaces are shown in fig. 25. As seen from fig. 25, the excited state of Hf atom (5d 6s ) inserts into Hj spontaneously to yield the Aj ground state of the Hf Hj molecule. The ground state of Hf has to surmount barriers for insertion into Hj. The smallest barrier is for the Aj state and it was computed as 17kcal/mol. 

Figures 26 and 27 show the computed potential energy surfaces of low and high spin states of ZrHj, respectively. The D excited state of the Zr atom inserts spontaneously into H2 to form the bent Aj state of the ZrH2 molecule. The high spin F state, to the contrary, needs to surpass barriers for insertion into H2.

Figure 5.16 shows the computed potential surfaces as contour plots in the (/ , ri) plane at r 2 = 1.37 A. Location (bond angle) and energy of the X " AijA conical intersection seam is shown in Fig. 5.17. Table 5.1 lists structure and energy parameters of the computed potential energy surfaces. The neutral ground state has an equilibrium bond length of 1.20 A and bond angle of 133° (1.196 A and 134.3° in the multireference configuration...

COMPUTED POTENTIAL ENERGY SURFACES FOR CHEMICAL REACTIONS... 

An analytic representation of the computed potential energy surface for reaction (6) was used in trajectory calculations which are discussed by Jaffe, Pattengill, and Schwenke in another article in this volume. 

---