J. Phys. Chem. 1993,97, 44014412
4407
Systematic Determination of Intersections of Potential Energy Surfaces Using a Lagrange Multiplier Constrained Procedure David R. Yarkony’ Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218 Received: December 7, I992
Two nonrelativistic Born-Oppenheimer potential energy surfaces of distinct spacespin symmetry intersect on a surface of dimension N - 1 where N i s the number of internal nuclear degrees of freedom. Characterization of this entire surface can be quite costly. An algorithm, employing multiconfiguration self-consistent-field (MCSCF)/configuration interaction (CI) wave functions and analytic gradient techniques, is presented which avoids the determination of the full N - 1 dimensional surface, while directly locating portions of the crossing surface that are energetically important. The algorithm is based on the minimization of the Lagrangian function LfJ(R,Xo,X) = E I ( R )+ ho[E,(R) - EJ(R)] ZE1X&(R) where Ck(R) is any geometrical equality constraint such as R K L-~a& = 0 or R K L-~R,& = 0, RKL= IRK- RL~,and XO and X are Lagrange multipliers. The efficacy of this algorithm is demonstrated using a simple MCSCF/first-order CI description of the spinforbidden reaction CH(X211) N2(X12,+) H C N ( X ’ V ) + N(4S). Sections of the crossing surface and the interstate spin-orbit couplings for the 4A” and 2A” potential energy surfaces are reported in the vicinity of a minimum energy crossing point.
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1. Introduction
Electronically nonadiabatic processes involve motion on more than one Born-Oppenheimer potential energy surface. In order to characterize the electronic structure aspects of such a process, it is necessary to determine the regions of nuclear coordinate space for which the surfaces in question are in close proximity. Such regions are characterized by conical intersections and avoided intersections. This work is concerned exclusively with potential energy surfaces determined from the nonrelativistic electronic Schrijdinger equation. In this case, it is useful to further classifysurface intersectionsas allowed intersections, intersections of two potential energy surfaces of distinct space-spin symmetry, and actual intersections, intersections of two potential energy surfaces of the same space-spin symmetry permitted by the multidimensional noncrossing ru1e.I For allowed intersections, the set of all intersection points is a surface of dimension N - 1, where N is the number of internal nuclear degrees of freedom. For actual intersections, the set of all intersection points is a surface of dimension N - 2. The lowest energy point on either of these surfaces is referred to as the minimum energy crossing point. Avoided intersections occur when for a given set of N 1 or N - 2 fixed parameters the minimum separation of two surfaces is not zero (see section 2); that is, crossings are permitted but not guaranteed. This work will focus on the determination of allowed crossings of states of different spin multiplicity. In addition,some discussion of an algorithm to treat avoided crossings will be also provided. The above discussion has assumed that the potential energy surfaces in question are determined from the nonrelativisticBornOppenheimer electronic Schrijdinger equation. Nonadiabatic transitions between surfaces of distinct spin multiplicity would then be induced by the spin-orbit interaction. This approach is standard and quite effective for the treatment of spin-forbidden processes involving low atomic number atoms.2.3 If the spinorbit interaction is incorporated into the Hamiltonian used to define the potential energy surfaces, conical intersections of the same dimension as those discussed above are obtained provided the system contains an even number of electron^.^ The situation
* Supported in part by DOE-BES Grant DE-FG02-91ER14189 and AFOSR Grant F49620-93-1-0067.
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is more complicated in odd electron systems. In this case, as discussed by Mead,4s5 the dimensionality of the surface of intersection is in general4N - 5 or N - 3 for planar systems,s and the treatment of the molecular Aharonov-Bohm effect associated with a conical intersectionl~~-~ is more complicated. This aspect of a conical intersection in odd electron systems is beyond the scope of the present work. The large dimensionality,N - 1, of an allowed crossing surface makes its determinationcostly and timeconsuming. Under certain circumstances, the requisite work can be reduced considerably by focusing attention on the minimum energy crossing point.* Such favorable circumstances are illustrated in parts a and b of Figure 1, which depict subclassesof spin-forbidden predissociation and spin-forbiddenbimolecular reactions for which the minimum energy crossing point represents the barrier to the spin-forbidden process. In these cases, the determination of the minimum energy crossing point provides key information concerning the feasibility of the process at hand. Algorithms for determining the minimum energy crossing point directly, that is, without prior determination of the crossing surface itself, have been discussed previously.9-l I Figure ICillustrates the situation motivating the present work. This figure depicts a spin-forbiddenbimolecular reaction in which much more of the crossing surface is energetically accessible. To address this situation, a systematic procedure for determining portions of the crossing surface will be presented. The algorithm is motivated by the desire to (i) avoid determining the full N 1 dimensional surface while (ii) locating those portions of the crossing surface that are energetically accessible. To accomplish this, points on the surface will be directly located-again, without prior determination of the crossing surface itself-subject to “arbitrary” equality constraints on some of the bond distances and/or bond angles, while the remaining geometrical parameters are optimized to minimize the energy of the crossing point. This constrained optimization procedure represents an extension of our previously reported’ method for determining the minimum energy crossing point on the crossing surface. The problem of geometry optimization, that is, the determination of stationary points, saddle points, and minima, on a single potential energy surface, has a long history in quantum chemistry and has been the subject of recent review^.'^,^^ The question of constrained minimizations on a single potential energy surface
0 1993 American Chemical Society 0022-3654/93/2091-4401~04.0~/0
Yarkony
4408 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993
Born-Oppenheimer potential energy surface, and, for example, the ith distance constraint is given by
CJR) = R ~ ~ a: ( ~ ) ~ (2.2a) or
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(2.2b) = R&fN(i)2 - RKL(i) 2 with R M= RM~RMN, ~ RMN= RM-RN,and RMe (XM,YM,ZM), locating the Mth nucleus of NT nuclei. Similar considerations will apply to other constraints such as those which restrict angles, but to simplify the presentation, direct reference will be made only to the constraints given in eq 2.2, which are expected to be the most useful in practical applications (see section 3). Minimizing eq 2.1 with respect to R, b,and X gives, at second order, the following Newton-Raphson equations
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ABC+D
AB+CD
MECP
MIN
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ABCD
ABC+D
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A
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AB+CD
MIN
MECP
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ABC+D
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Figure 1. (a, top) Schematic representation of a spin-forbidden predissociation for which the minimum energy crossing point (shaded area) constitutes the barrier for the reaction. (b, middle) Schematic representation of a spin-forbidden bimolecular reaction for which the minimum energy crossing point (shaded area) constitutesthe barrier for thereaction. (c, bottom) Schematic representation of a spin-forbidden bimolecular reaction for which an appreciable region of the crossing hypersurface is energetically accessible.
has also been considered, particularly in the context of molecular mechanics proceduresI4-16 using both Lagrange multiplier14J5 and penalty function16 approaches and has very recently been considered by Truhlar and co-workers in the context of small molecule chemistry." In the method of Truhlar and co-workers, a projection operator approach is used to constrain the minimization procedure. The present work provides the first report of an approach for performing constrained minimizations .on an allowed crossing surface. Here Lagrange multiplierswill be used to both constrain the minimization to the crossing surface and impose the geometrical constraints. The advantages of this approach are described in section 2, which reports the method and its implementation. In section 3, the use and efficacy of the algorithm are illustrated with the help of representative calculations in the vicinity of a minimum energy crossing point for the reaction CH(X2II) Nz(XI&+) HCN(XIZ+) + N(4S). Section 4 summarizes and indicates directions for future investigation.
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2. Theoretical Approach A. Constrained Minimization Procedure. The algorithm presented here determines the minimum energy point on the crossing surface subject to a set of geometrical constraints of the form Ci(R) = 0, i = 1, M,where R are the nuclear coordinates. This is accomplished by minimizing the Lagrangian function9 L'J(R,b,N
where 6R = R' - R, SAj = A) - Aj, the energy gradient = (d/dWa)EI(R), the energy difference gradient &
