The diagonal Born±Oppenheimer correction to molecular dynamical properties

Transcription

1 6 January Chemical Physics Letters 333 () 459±464 The diagonal Born±Oenheimer correction to molecular dynamical roerties Sohya Garashchuk a, *, John C. Light a, Vitaly A. Rassolov b a James Franck Institute, University of Chicago, Chicago, IL 6637, USA b Deartment of Chemistry, Northwestern University, Evanston, IL 68, USA Received 7 Setember ; in nal form 8 October Abstract We examine the e ect of the diagonal Born±Oenheimer correction on dynamics in two simle systems ± the Hooke's atom in an external harmonic otential and the collinear hydrogen exchange reaction. The transmission robability for the Hooke's atom, calculated within the Born±Oenheimer aroximation, is simly shifted in energy with resect to the exact result, and this is corrected by the diagonal adiabatic contribution. The reaction robability for the H 3 system re ects the fact, that the diagonal Born±Oenheimer correction raises the barrier to the reaction by aroximately 7 cm. Ó Elsevier Science B.V. All rights reserved.. Introduction The increase in accuracy and e ciency of ab initio quantum chemistry methods and the ability to study non-adiabatic molecular dynamics of small molecules brings attention to small corrections to the Born±Oenheimer electronic otential surface. The state-of-the-art electronic structure calculations aroach the accuracy of a few wave numbers []. The agreement of the best sectroscoic calculations with exeriment can be better than a wave number [,3]. Thus, the relativistic, adiabatic and non-adiabatic corrections become imortant. The non-adiabatic e ect of `electrons following nuclei' that does not involve transitions between the otential surfaces is often * Corresonding author. Fax: address: sgarashc@midway.uchicago.edu (S. Garashchuk). simulated by using atomic rather than nuclear masses for some of the degrees of freedom or even some intermediate between the atomic and nuclear masses [4,5]. The adiabatic diagonal Born±Oenheimer correction (DBOC) is generally neglected. A few studies show that the e ect of DBOC is, indeed, small for the equilibrium roerties of small molecules, such as bond distances and fundamental frequencies. Several small molecules, such as H and He were examined by Handy and coworkers [6,7]. The e ect of DBOC was found to be small (change in bond length on the order of 4 ± 5 bohr) for HF, N,F, but a DBOC was essential to obtain an accurate dissociation energy of H. Moss [8] found the adiabatic correction to the bond length of H to be about.3%, while the non-adiabatic correction was three orders of magnitude smaller. For the equilibrium con guration, the DBOC has weak deendence on the nuclear osition (for the harmonic oscillatorlike otential electronic surface it is constant). 9-64//$ - see front matter Ó Elsevier Science B.V. All rights reserved. PII: S ( ) 97-5

2 46 S. Garashchuk et al. / Chemical Physics Letters 333 () 459±464 However, one might exect more ronounced DBOC e ects for reactive systems, where the electronic otential energy surface is oorly aroximated by harmonic oscillator solutions in the interaction region of the otential surface. For examle, the DBOC lowers the barrier to linearity of water by 7 cm [9] and it raises the barrier for the reaction F H! FH H by 7 cm [7]. The barrier to linearity of hydrogen sul de is also raised by about 7 cm []. In this Letter, we examine the DBOC e ects on dynamics of two simle systems by nding reaction robabilities. In Section, we nd the transmission robability for an exactly soluble system, the Hooke's atom in the arabolic external otential where the DBOC, indeed, comensates for BO aroximation errors in the rst order. Then, we look at the state-resolved reaction robabilities of the collinear hydrogen exchange reaction. The DBOC is calculated within a comact single-term formulation [6,]. We use an unrestricted Hartree±Fock (UHF) electronic wave function in our calculation. We also use the Mùller±Plesset UMP sin-unrestricted wave function to investigate the in uence of the single determinant aroximation on DBOC. The correction is added to the LSTH [±4] electronic otential surface for H 3. Aart from the overall shift of the otential surface by 68 cm, the barrier is raised by an additional 7 cm, which is re ected in the reaction robability. The e ect of using atomic instead of nuclear mass, which is assumed to correct for the `electrons following nuclear' e ect, with or without DBOC is negligible, shifting the sectrum by aroximately.5 cm. Section 4 concludes.. Hooke's atom in the external arabolic otential The standard way to solve the Schrodinger equation for a system with the electronic and nuclear degrees of freedom is to use the Born±Oenheimer aroximation to the total wave function jwi. jwi is reresented as a roduct jwi ˆjvij/i, where j/i ˆ/ r; R is the eigenfunction in the electronic coordinate r of the electronic art of the Schrodinger equation, solved for xed values of the nuclear coordinate R. The energy eigenvalue V R is the electronic otential energy surface (PES) and R aears as a arameter in / r; R. jvi ˆv R is the solution of the nuclear art of the Schrodinger equation in the otential V R. The general form of the diagonal BO correction [], which is indeendent of the center of mass motion, is V c R ˆ h h/j X l M l r Rl j/i; where l is the index for nuclei and M l are the masses of nuclei. The nuclear masses also aear in the kinetic energy art of the nuclear Hamiltonian. First, let us look at a simle soluble roblem ± the one-dimensional Hooke's atom, which is a system of a roton and an electron bound by a harmonic otential, in an external arabolic otential. The coordinate of the roton is X and the mass of a roton is M. The coordinate of the electron is x and the electronic mass is. We use h ˆ below. The Hamiltonian is ^H ˆ M o ox o ox k X x g X g ex ; where the roton±electron interaction is controlled by the arameter k and the arameters of interaction with the external otential is g for a roton, and g e for electron. Solving the model exactly in the normal mode coordinates y; Y : y ˆ cg x k a g X ; 3 Y ˆ k a g x cg X ; s k a b ˆ c 4 a b ; 4 a ˆ k g e ; b ˆ k g =M; c ˆ k= M ; q g ˆ k a c gives a decouled Hamiltonian ^H ˆ 5 o oy o oy k Y k y : 6

3 S. Garashchuk et al. / Chemical Physics Letters 333 () 459± Under the conditions k >, g e P, g P, g e g 6ˆ the motion is bounded in y, but is unbounded in Y. The ground state of the harmonic oscillator in the y coordinate gives the energy of the to of the barrier in Y V ˆ k =: 7 The otential in Y coordinate becomes V ˆ V w Y = with the barrier frequency w ˆ k : 8 Solving the same roblem in the BO aroximation we have the barrier height for the electronic ground state V BO ˆ k g e = 9 and the barrier frequency s w BO ˆ g kg e M k g : e The DBOC for the electronic ground state is V c ˆ o M h/ j iˆvbo ox j/ M : Exanding in the rst order of =M for g e ˆ we have 4V ˆ k k k=m; w ˆ k g =M g =M for the exact solution and 4 V BO ˆ k; w BO ˆ g =M: 3 The corrected barrier height within the BO aroximation is V BO V c k ˆ ; 4 M which is in agreement within the rst-order in =M with the exact result k V M 4g : 5 8M k The transmission robability for the arabolic barrier is [5] N E ˆ ex ; 6 where ˆ E V =w: 7 Although V agrees with V BO V c and w agrees with w BO within the rst-order of =M, arameters for the exact and the BO corrected cases are di erent by c ˆ E k M : g This is the e ect of the overall motion of the roton and electron. The discreancy is zero when E is equal to the zero-oint energy of the relative (vibrational) motion and it is large for energies that are either very high or very low comared to the barrier height. The absolute reaction robability is little a ected by this error due to its functional deendence on (Eq. (6)), which may be a general feature of most chemical systems. We also note, that if the nuclear mass M is relaced with the atomic mass M in the BO corrected exressions (9) and (), the resulting c does agree with the exact within O =M, which justi es the usage of the atomic mass rather than nuclear mass in molecular dynamics calculations. Nevertheless, the e ect of using atomic rather than nuclear mass is small comared to the e ect of the DBOC itself. Fig. comares the exact result with the BO aroximation exact BO aroximation BO with DBOC Energy, hartree Fig.. Transmission robability, N E, for the Hooke's atom in a arabolic otential calculated exactly, within BO aroximation with and without the DBOC.

4 46 S. Garashchuk et al. / Chemical Physics Letters 333 () 459±464 BO ˆ E V BO =w and with the BO aroximation with the DBOC c ˆ E V BO V c =w for the value of arameters M ˆ 98 a.u., k ˆ :38, g ˆ :647 and g e ˆ, which aroximately corresonds to the H 3 system using nuclear mass. For given arameters the e ect of using atomic instead of nuclear mass is small: the maximal di erence between the exact and the Born± Oenheimer robabilities without the DBOC correction is.3 hartree using nuclear or atomic mass, while the di erence between exact N E and the Born±Oenheimer robability with the DBOC correction is :4 4 if using nuclear mass and :7 5 hartree if using atomic mass. For our model with quadratic otentials the DBOC is indeendent of the nuclear osition. So the Born±Oenheimer aroximation results in an overall shift of the barrier height, that is very accurately comensated by the diagonal correction in the rst order in M. For chemical systems the DBOC does deend on the osition of nuclei, even more so for systems far from equilibrium as demonstrated in the next section. 3. State-to-state reaction robability for collinear H 3 system In order to see the magnitude of the e ect of the DBOC on dynamics, we calculate the state-to-state transition robabilities of the collinear hydrogen reaction for the two lowest vibrational states. We use the wave acket correlation formulation to calculate reaction robabilities for! and! reactive transitions. The incoming reactant and outgoing roduct wave ackets are de ned in the asymtotic region of the otential surface in Jacobi coordinates. The wave ackets are de ned as the direct roducts of Gaussians in the translational coordinate and the eigenstate of the Morse oscillator in the vibrational coordinate. The arameters of the reactant and roduct translational wave ackets, U R ˆex a R R õ R R, in the aroriate asymtotic region are a ˆ 6:, R ˆ 6: and ˆ 7: in atomic units. The wave acket arameters are the same for all calculations with and without the correction and using: (a) atomic and (b) nuclear masses. The robabilities are obtained from the ratio of the energy sectrum of the reactant/roduct wave acket correlation function to the sectrum of the reactant/reactant correlation function [6]. For the electronic structure calculations, we use a large 7s5d basis, with the innermost s function being a contraction of ve rimitives, and the rest of the functions uncontracted. The values of the DBOC changed only marginally with the addition of more rimitives, so our basis is essentially comlete within the accuracy of this Letter. The electronic structure calculations were used only to determine the DBOC which was added to the LSTH otential energy surface. We have used a modi ed version of Q-Chem software ackage for the DBOC calculation [7]. A larger error in the DBOC comes from the single determinant aroximation to the electronic wave function. The DBOC has a singularity in the asymtotic H+H region at.36 bohr (RHF! UHF instability) in H. This is an artifact of the single determinant aroximation. Fortunately, we found this discontinuity to be high enough on the otential surface to have a negligible e ect on the two lowest transitions studied in this Letter. The investigation of the single determinant aroximation was done by comaring the DBOC calculated with UHF wave function and erturbatively to the leading order with the UMP wave function [8]. They agree within 3±4% in the asymtotic region and within ±% in the barrier region in the low-energy region of the otential surface, shown in Fig.. Overall, we comared reaction robabilities for three di erent DBOCs. The rst one is the DBOC within UHF theory, calculated for the bond distances :66 < X < 8:9 bohr, :66 < Y < 4:5 bohr and set to zero elsewhere. Note, that this DBOC contains the singularity region of PES. The second calculation is DBOC obtained from the UMP wave function calculated for :66 < X < 8:9 bohr, :66 < Y < :36 bohr and set to zero elsewhere. This calculation does not contain the afore-mentioned singularity. The third calculation uses DBOC of the second calculation, but with the singularity smoothed over and with the correction

5 S. Garashchuk et al. / Chemical Physics Letters 333 () 459± barrier, UHF barrier, UMP asym., UHF asym., UMP.8 nuclear mass, without DBOC atomic mass, without DBOC nuclear mass, DBOC (UMP) 4 Correction, /cm Distance, bohr Energy, ev Fig.. DBOC for the asymtotic and barrier regions (X ˆ 8:5 bohr and X ˆ Y in bond coordinates, resectively) calculated with UHF and UMP wave functions. extraolated outside the calculated region. All three corrections are added to the same LSTH otential surface. The resulting three reaction robabilities are insensitive to the di erences of the corrections as shown on Fig. 3. This shows that the wave acket never reaches the singularity region, which is quite high in energy, and that the UHF wave function is adequate for DBOC in the low energy region of PES. The e ect of DBOC on dynamics is de ned by the changes in the interaction region relative to the Fig. 4. Reaction robability for the collinear H 3 system for! transition calculated within BO aroximation without DBOC using nuclear and atomic masses and with DBOC using nuclear mass. The robability obtained with the DBOC is shifted by about 7 cm. changes in the asymtotic region. The BO correction raises the minimum of the otential surface by 68 cm. This is equivalent to the energy shift by cm for H and is consistent with the results of Kolos and Rychlewski [9] 4.6 cm for the distance of R ˆ :4 bohr. The barrier is raised by an additional 7 cm. The relative shift of the barrier height determines the main change in the! reaction robability shown on Fig. 4. Fig. 5.8 without DBOC UHF DBOC UMP DBOC UMP DBOC smoothed nuclear mass, without DBOC nuclear mass, DBOC (UMP) Energy, ev Fig. 3. Reaction robability for the collinear H 3 system for! transition calculated with UHF, UMP and with the smoothed UMP corrections using nuclear mass Energy, ev Fig. 5. Reaction robability for the collinear H 3 system for! transition calculated with and without DBOC using nuclear mass.

6 464 S. Garashchuk et al. / Chemical Physics Letters 333 () 459±464 shows the reaction robability of! transmission for which the di erence between the BO robability and the corrected robability is more comlicated than a mere shift. The Born±Oenheimer aroximation imlies that the nuclear masses are used. In ractice, atomic masses are used to mimic the `electron following the nucleus' e ect. We reeated the calculations using atomic masses instead of nuclear masses and found a very small shift of the sectrum of about ).5 cm, which is an order of magnitude smaller than the DBOC e ect. This is shown on Fig. 4 for calculations without the DBOC, and the curves obtained with atomic and with nuclear masses are indistinguishable. 4. Summary The diagonal Born±Oenheimer correction to electronic otential surfaces becomes imortant in the context of high accuracy theoretical and exerimental methods of hysical chemistry. It can be readily incororated into ab initio calculation of the electronic otential surfaces. Though the DBOC has small e ect on the equilibrium roerties of molecules, the DBOC in the barrier region has a noticeable e ect on reaction robabilities, as was demonstrated for the examle of collinear H 3. The DBOC does shift the barrier height by a small but noticeable amount on the order of cm for H 3. The DBOC was calculated erturbatively using UHF and UMP wave functions. We conclude that a single determinant UHF wave function is adequate for describing the DBOC in the barrier region. The e ect of the mass substitution (atomic versus nuclear) on the reaction robabilities is negligible at low energies, as follows from the analysis of both systems, the harmonic model and the collinear H 3. Acknowledgements The authors acknowledge the suort in art by a grant from the Deartment of Energy, DE- FG-87ER3679 and thank the chemistry division of the NSF for suort. References [] Y.S. Mark Wu, A. Kuermann, J.B. Anderson, Phys. Chem. Chem. Phys. (998) 99. [] H. Partridge, D.W. Schwenke, J. Chem. Phys. 6 (997) 468. [3] B.J. McCall, T. Oka, J. Chem. Phys. 3 () 34. [4] R.E. Moss, D. Joling, Chem. Phys. Lett. 6 (996) 377. [5] R. Jaquet, Chem. Phys. Lett. 3 (999) 7. [6] N.C. Handy, A.M. Lee, Chem. Phys. Lett. 5 (996) 45. [7] A.G. Ioannou, R.D. Amos, N.C. Handy, Chem. Phys. Lett. 5 (996) 5. [8] R.E. Moss, Mol. Phys. 97 (999) 3. [9] G. Tarczay, A.G. Csaszar, W. Kloer, W.D. Allen, H.F. Schaefer, J. Chem. Phys. (999) 97. [] G. Tarczay, A.G. Csaszar, M.L. Leininger, W. Kloer, Chem. Phys. Lett. 3 () 9. [] W. Kutzelnigg, Mol. Phys. 9 (996) 99. [] P. Siegban, B. Liu, J. Chem. Phys. 68 (978) 457. [3] D.G. Truhlar, C.J. Horowitz, J. Chem. Phys. 68 (978) 466. [4] W. Kolos, L. Wolniewicz, J. Chem. Phys. 43 (965) 49. [5] T. Seideman, W.H. Miller, J. Chem. Phys. 95 (99) 768. [6] S. Garashchuk, D.J. Tannor, J. Chem. Phys. 9 (998) 38. [7] C.A. White, J. Kong, D.R. Maurice, T.R. Adams, J. Baker, M. Challacombe, E. Schwegler, J.P. Dombroski, C. Ochsenfeld, M. Oumi, et al., Q-Chem, Inc., Pittsburgh, PA (998), version.. [8] V. Rassolov, in rearation. [9] W. Kolos, J. Rychlewski, J. Chem. Phys. 98 (993) 396.