High Accuracy Rotation–Vibration Calculations on Small Molecules Jonathan Tennyson Department of Physics and Astronomy, University College London, London, UK 1 INTRODUCTION comprehensive understanding requires the knowledge of many millions, perhaps even billions, of individual transi- High-resolution spectroscopy measures the transitions bet- tions (Tennyson et al. 2007). It is not practical to measure ween energy levels with high accuracy; typically, uncer- this much data in the laboratory and therefore a more real- tainties are in the region of 1 part in 108. Although it is istic approach is the development of an accurate theoretical possible, under favorable circumstances, to obtain this sort model, benchmarked against experiment. of accuracy by fitting effective Hamiltonians to observed The incompleteness of most experimental datasets means spectra (see Bauder 2011: Fundamentals of Rotational that calculations are useful for computing other properties Spectroscopy, this handbook), such ultrahigh accuracy is that can be associated with spectra. The partition functions largely beyond the capabilities of purely ab initio proce- and the variety of thermodynamic properties that are linked dures. Given this, it is appropriate to address the question to this (Martin et al. 1991) are notable among these. Again, of why it is useful to calculate the spectra of molecules ab calculations are particularly useful for estimating these initio (Tennyson 1992). quantities at high temperature (e.g., Neale and Tennyson The concept of the potential energy surfaces, which in 1995). turn is based on the Born–Oppenheimer approximation, Other more fundamental reasons for calculating spectra underpins nearly all of gas-phase chemical physics. The include the search for unusual features, such as clustering of original motivation for calculating spectra was to provide energy levels (Jensen 2000) or quantum monodromy (Child stringent tests of potential energy surfaces. As discussed et al. 1999), or to link with semiclassical analysis. These in Section 5, this procedure has now been turned round reasons are not discussed here. and spectra can be used to construct or more commonly Many of the above reasons relate closely to working refine potential energy surfaces. The fact that spectra can with measurements but the results from calculations can be recorded with high accuracy does not necessarily mean be important in their own right. For most practical appli- that they are easy to interpret. The second motivation for cations of spectroscopy, such as those involved in remote first-principles calculation of spectra is to make predictions sensing or radiative transfer modeling, information on tran- before any experiment and to aid with the assignment sition intensities is as important as that on line positions. process afterward. Accurate absolute intensity measurements are difficult; they A third reason is the analysis of spectra or, indeed, can usually only be made in situations where it is possible radiation transport models at high temperatures. The to get a well thermalized, known column of the sample and rotation–vibration spectra of even rather small molecules become very complicated at elevated temperatures. A even then certain experiments simply do not yield this infor- mation. For small molecules, ab initio theoretical techniques Handbook of High-resolution Spectroscopy. Edited by Martin Quack are advanced sufficiently that the intensities they produce and Fred´ eric´ Merkt. 2011 John Wiley & Sons, Ltd. may often be more trustworthy than laboratory ones, even ISBN: 978-0-470-74959-3. when they are available. 552 High Accuracy Rotation–Vibration Calculations on Small Molecules 2 SEPARATING ELECTRONIC, As the notation shows, the nuclear motion wave function, nuc VIBRATIONAL, ROTATIONAL, ψ , is a function only of the nuclear coordinates and the electronic motion wave function, ψelec, depends explicitly AND TRANSLATIONAL MOTION on the electronic coordinates and parametrically on the nuclear motion coordinates. That is, there is a different It is straightforward to write down the Schrodinger¨ equation electronic wave function for each nuclear configuration. In governing the motion of a molecule containing N atoms practice, this wave function depends only on the internal or each with charge ZI and mass MI and n electrons, each relative nuclear coordinates, which are defined below. − with charge e and mass me. The associated nonrelativistic Within the Born–Oppenheimer approximation, the elec- Hamiltonian is tronic wave function is obtained as the solution of the Schrodinger¨ equation given by: N 2 2 n N n eZ Hˆ =− ∇2 − ∇2 − I I i elec elec elec 2MI 2me |RI − ri| ˆ + + = I=1 i=1 I=1 i=1 (Te VNe Vee)ψ E ψ (3) N n 2 ZI ZJ e The electronic energy, Eelec, obtained from this equation − − /(4πε0) |RI − RJ | |ri − rj | depends parametrically on the relative nuclear coordinates, I=1 J<I i=1 j<i Q; it is used in the nuclear motion Schrodinger¨ equation: = ˆ + ˆ + + + TN Te VNe VNN Vee (1) ˆ nuc nuc TN + V(Q) ψ = Eψ (4) where the terms denote the nuclear motion kinetic energy operator, Tˆ , the electronic motion kinetic energy operator, N where E is the total energy for the state under consideration. Tˆ , the potential due to the coulombic attraction between e A very important part of equation (4) is the potential energy the electrons and the nuclei, V , the potential due to Ne surface V which is defined as: the coulombic repulsion between the nuclei, VNN and the potential due to the coulombic repulsion between the elec- BO elec V(Q) = V (Q) = E (Q) + VNN(Q) (5) trons, Vee. The coordinates used to represent the Hamil- tonian of equation (1), RI for the nuclei and ri for the electrons, are all defined relative to some arbitrary origin. This potential is independent of the mass of the nuclei This Hamiltonian has 3(N + n) degrees of freedom. and therefore is the same for all isotopologues of a Though it may be straightforward to write down the particular molecule. This is an important result of the Born– Hamiltonian given by equation (1), there are a number Oppenheimer approximation and one which can be used to of steps that need to be taken before tractable solutions quantify its reliability. can be found to the related Schrodinger¨ Equation. These The basis of the Born–Oppenheimer approximation is steps can be characterized as separating the nuclear and that the electrons move rapidly compared to the nuclei electronic motions, removing the translational motion of the because me is very much less than MI . This approximation whole system and, less crucially, distinguishing between the is worst for the lightest nucleus, hydrogen, where MH is still vibrational and rotational nuclear motion. Although there 1836 me. There are also serious problems with the Born– are different methods available for studying molecules that Oppenheimer approximation when two electronic potential skip each of these steps, none of them have been generally energy surfaces come very close to each other or even used to study the rotation–vibration states of chemically cross, because under this circumstance, the separability of bound molecules; these methods are not discussed here. the electronic and nuclear wave functions, equation (2), can fail badly. The Born–Oppenheimer approximation actually involves 2.1 The Born–Oppenheimer Approximation two approximations: the separation of the electronic and nuclear wave functions, equation (2), and neglect of the term where the nuclear kinetic energy operator, Tˆ acts on The first standard step, the separation of electronic and N the electronic wave function, ψelec. It is reasonably straight- nuclear motion, is based upon what is known as the Born– forward to improve on the Born–Oppenheimer approxima- Oppenheimer approximation. This approximation starts by tion by allowing for the case where Tˆ acts on ψelec.The characterizing the total wave function of the system as a N simplest way of writing this correction, W ad,is: product of electronic and nuclear wave functions: elec nuc W ad(Q) = ψelec|Tˆ |ψelec (6) Ψ = ψ (ri; RI )ψ (RI ) (2) N High Accuracy Rotation–Vibration Calculations on Small Molecules 553 where the integration implied by the Dirac notation is over those lying in the electronic continuum. Formally the all electronic coordinates. This is the Born–Oppenheimer nonadiabatic correction can therefore be written: diagonal correction (BODC) and adding it to the potential nonad = elec| ˆ | elec of equation (5) gives the adiabatic approximation. It is W (Q) ψ0 TN ψn (9) important to note that the BODC is mass dependent n>0 and therefore there is a new effective potential for each The infinite sum over electronic states makes working isotopologue in the adiabatic approximation. with this form of the wave function very difficult, and For high-accuracy work, it is therefore often desirable to full treatments of the nonadiabatic corrections are rare. In define a more general, effective, potential energy function practice, the sum is often truncated after two or three terms which can be written as follows: in the important case of electronic states which lie near to each other. Schwenke (2001) has given a treatment based W eff(Q) = V BO(Q) + V rel(Q) + W ad(Q) (7) on performing the full sum over the finite set of states given by a restricted (Hartree–Fock) treatment of the electronic eff W is the sum of the mass-independent Born– problem. rel Oppenheimer potential plus relativistic correction, V , More pragmatic treatments of the nonadiabatic problem which is discussed below, and the mass-dependent adi- are usually employed. The starting point for these is the abatic correction. Note that the mass dependence of the observation that the nonadiabatic corrections are usually adiabatic correction is implicitly included in the nuclear ˆ best modeled by corrections to the nuclear kinetic energy kinetic energy operator, TN , in equation (6). ˆ ˆ operator, TN , rather than the potential.