Article pubs.acs.org/JPCA
Canonical Potentials and Spectra within the Born−Oppenheimer Approximation Jay R. Walton Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, United States
Luis A. Rivera-Rivera, Robert R. Lucchese, and John W. Bevan* Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, United States ABSTRACT: A generalized formulation of canonical transformations and spectra are used to investigate the concept of a canonical potential strictly within the Born−Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intermolecular potentials in H2, HD, D2, HeH+, and LiH are used to rigorously evaluate such transformations. The corresponding potentials are generated explicitly using parameters calculated with algebraic functions from that of the single canonical potential of the simplest molecule, H2+. The efficacy of this approach is further tested by direct comparison of the predicted eigenvalues of all vibrational states in the selected molecular systems considered with the corresponding most accurately known Born−Oppenheimer eigenvalues currently available. Deviations are demonstrated to be less than 2 cm−1 for all vibrational states in H2, HD, D2, HeH+, and LiH, with an average standard deviation of 0.27 cm−1 for the 87 states considered. The implications of these results for molecular quantum chemistry are discussed.
I. INTRODUCTION It is recognized that forces between atoms directly or indirectly influence almost all molecular phenomena.1 Knowledge of accurate interatomic potential energy surfaces are fundamental to the characterization and prediction of many properties in all states of matter whether the forces are between atoms or unreacting molecules and fragments of molecules.2−9 A coherent representation for ubiquitous radially dependent two body pairwise interatomic interaction potentials remains elusive.10−12 Functional forms to describe interatomic potentials such as Mie,13 Lennard-Jones,14 and Morse15 potentials have now proliferated to over 100 different algebraic functions16 developed assuming the existence of a general potential function, different only for various diatomic molecules through involvement of two, three, or more adjustable parameters. However, the search for a unique reduced17 or universal10 potential for accurately representing a wide range of such molecules continues unabated. Recently, we introduced formulations for accurately generating equilibrium dissociation energies, De, as well as explicit force-based transformations18 to a canonical potential for both diatomic and two-body intermolecular interactions using parameters calculated explicitly from algebraic functions.19 Different classes of representative ground electronic state pairwise interatomic interactions were referenced to a chosen canonical potential illustrating application of such transformations. Specifically, accurately determined potentials of the diatomic molecules H2, H2+, HF, LiH, argon dimer, and one-dimensional dissociative coordinates in intermolecular Ar− HBr, OC−HF, and OC−Cl2 were investigated throughout their © XXXX American Chemical Society
ground state bound potentials. The results indicated that an explicit transformation developed specifically for the Born− Oppenheimer potential20 of H221 can be applied to the other selected two-body molecular potentials used to generate such a corresponding canonical potential. Such applications included interatomic interactions that had been only partially corrected for non-Born−Oppenheimer effects.22−25 That formulation18 was able to predict De energies by extrapolating the attractive force to zero for the systems investigated. Once transformed into this dimensionless form, the resulting potentials were all consistent within an accuracy of better than 0.4%; however, questions arise as to the origin of such deviations, including whether they are a consequence of the fundamental limitations of the canonical transformation approach or are due to a lack of sufficient corrections for non-Born−Oppenheimer effects in the systems investigated. We now study this generalized formulation of canonical transformations19 and spectra to investigate the concept of a canonical potential strictly within the Born−Oppenheimer approximation.20 Specifically, from the single canonical potential of the simplest molecule, H2+, alone, we generate the potentials of a set of different molecules explicitly using parameters calculated with algebraic functions. This set was chosen to include the most accurately available ground electronic state pairwise interaction potentials in H2, HD, D2,21,26,27 HeH+,28,29 and LiH.30 The efficacy of the canonical Received: April 27, 2015 Revised: June 2, 2015
A
DOI: 10.1021/acs.jpca.5b04008 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A approach is then stringently tested by direct comparison of the predicted eigenvalues of all vibrational states in the ground electronic state bound potentials of the molecular systems considered with those of the corresponding most accurately known Born−Oppenheimer eigenvalues currently available. Such deviations will be demonstrated to be Re F(R), respectively. Then define the two dyadic sequences Re < ... < Raj−1 < Raj < ... and 0 < ... < Rrj < Rrj−1 < ... < Re by F(R ja)
Rai )/(γ(Raj
(3) B
DOI: 10.1021/acs.jpca.5b04008 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A dimensional potential V̆ (R) for the reference molecule, H2+, that take the forms
with the power n(l−1)il chosen so that the tail distribution interpolates V(R) at the two points Rai(l)−1il(γ) and Rail−1.
V a(R ; ej , γ ) = V (R e) + Aeaj {V̆ (R̆ e + Aeaj (R − R e)) − V̆ (R̆ e)}
for R e < R < R eaj(γ ) a
nil =
(10)
(11)
V r(R ; ej , γ ) = V (R e) + Aerj {V̆ (R̆ e + Aerj (R − R e)) − V̆ (R̆ e)} for R erj(γ ) < R < R e
(12)
r r V r(R ; ij , γ ) = V (R ir) + Aijr {V̆ (R̆ i + Aijr (R − R ir)) − V̆ (R̆ i )}
for R ijr(γ ) < R < R ir
log(R ia(l−1)il /R ial − 1)
(17)
iv. Eigenvalue Calculations. To solve the radial Schrödinger equation, the accurate and estimated potential for the considered molecules were interpolated using cubic spline. For all molecules, except for LiH, the radial grid was 0.15−10.0 Å. For LiH, the radial grid was 0.6−10.0 Å. The estimated potential of HeH+ and the accurate and estimated potentials for LiH were extrapolated on the repulsive wall using the exponential function V(R) = [a × exp(−b × R)] + c, where the constants a, b, and c were calculated using the last three points of the repulsive wall of the given potential. The accurate potential for LiH was also extrapolated on the attractive side using a Lennard-Jones tail with the parameters interpolated to the last two points of the attractive potential. The eigenvalues were then calculated using a modify Numerov−Cooley approach.31
a
V a(R ; ij , γ ) = V (R ia) + Aija {V̆ (R̆ i + Aija (R − R ia)) − V̆ (R̆ i )} for R ia < R < R ija(γ )
log(V (R ial − 1)/V (R ia(l−1)il))
(13)
where R eβj (γ ) = γR jβ + (1 − γ )R e , R ijβ(γ ) = γR jβ + (1 − γ )R iβ (14)
III. RESULTS AND DISCUSSION The accurately determined 15 bound vibrational eigenvalues within the Born−Oppenheimer approximation are given for H221 in column 2 of Table 1. The corresponding values
⎛ V (R β(γ )) − V (R ) ⎞ ⎛ V (R β(γ )) − V (R ) ⎞ ej e ⎟ ij e ⎟ β ⎜ A = Aeβj = ⎜ ⎜ V̆ (R̆ β) − V̆ (R̆ ) ⎟ ij ⎜ V̆ (R̆ β) − V̆ (R̆ ) ⎟ ⎝ ⎠ ⎝ ⎠ j e j i (15)
Table 1. Vibrational Eigenvalues for H2
In eqs 14 and 15, the superscript β correspond to the superscript a or r in eqs 10−13. It is important to note that eq 10 can be seen to be an elementary affine transformation of the portion of the reference potential curve V̆ (R) specified by R̆ e < R < R̆ j onto the portion of the potential curve V(R) specified by Re < R < Raej(γ). Specifically, this affine transformation merely maps the two points {(R̆ e,V̆ (R̆ e)),(R̆ j,V̆ (R̆ aj ))} on the reference potential curve onto the corresponding points {(Re,V(Re)),(Raej(γ),V(Raej(γ)))}. In the applications below, the parameter γ is chosen through the additional requirement that the affine transformation also interpolates the midpoints of the potential curve segments, that is, the point {((R̆ e + R̆ aj )/2),(V̆ ((R̆ e + R̆ aj )/2))} on the reference potential curve maps onto the corresponding point {((Re + Raej(γ))/2),(V((Re + Raej(γ))/2))} on the given potential curve. Similar comments apply to eqs 11−13. iii. Canonical Dimensional Potential Representations. The potential for a given molecule within the considered class is approximated to very high accuracy by the forward and reverse piecewise affine canonical transformation of the reference potential for H2+. To that end, for a given molecule, one constructs a set of internuclear separations {R̆ rik < ...
