Calculation of Converged Rovibrational Energies and Partition Function for Methane using Vibrational-Rotational Configuration Interaction

Transcription

1 February 2, 2004 Prepared for publcaton n J. Chem. Phys. Calculaton of Converged Rovbratonal Energes and Partton Functon for Methane usng Vbratonal-Rotatonal Confguraton Interacton Arndam Chakraborty, Donald G. Truhlar Department of Chemstry and Supercomputng Insttute, Unversty of Mnnesota, Mnneapols, MN Joel M. Bowman, and Stuart Carter Cherry L. Emerson Center of Scentfc Computaton and Department of Chemstry, Emory Unversty, Atlanta, GA Abstract. The rovbraton partton functon of CH 4 was calculated n the temperature range of K usng well-converged energy levels that were calculated by vbratonal-rotatonal confguraton nteracton usng the Watson Hamltonan for total angular momenta J = 0 50 and the MULTIMODE computer program. The confguraton state functons are products of ground-state occuped and vrtual modals obtaned usng the vbratonal self-consstent feld (VSCF) method. The Glbert and Jordan potental energy surface was used for the calculatons. The resultng partton functon was used to test the harmonc oscllator approxmaton and the separable-rotaton approxmaton. The harmonc oscllator, rgd-rotator approxmaton s n error by a factor of two at 300 K, but we also propose a separable-rotaton approxmaton that s accurate wthn 2% from 100 K to 1000 K.

2 2 I. INTRODUCTION For accurate calculatons of molecular energy levels, spectra, and thermochemstry, t s essental to take account of anharmoncty and the nteracton between rotaton and vbraton. The couplng between rotaton and vbraton s due to Corols and centrfugal terms. A revew of perturbaton methods to account for anharmoncty and rotaton-vbraton couplng was gven by Nelsen. 1 For a hghly symmetrc molecule lke methane the anharmoncty and rotaton-vbraton nteractons may be analyzed usng group theory. 2,3 Jahn used group theory and frst order perturbaton theory to treat rotaton-vbraton nteractons n methane n a seres of four papers. 4 7 Later, second order and thrd order perturbaton calculatons were reported by Shaffer et. al. 8,9 and by Hecht. 10,11 More recently Lee et. al. 12 and Wang et. al. 13 reported vbratonal perturbaton theory calculatons on methane usng an analytcal potental energy surface. Another approach to the computaton of rovbratonal levels of molecules s based on varatonal theory. The vbratonal self-consstent feld (VSCF) method s one such varatonal procedure. The VSCF procedure was extended by Carter et. al. 20,21 to study rovbratonal states by usng the Whtehead-Handy 22,23 mplementaton of the Watson 24 Hamltonan. However, the VSCF method s not quanttatvely accurate. A more accurate, systematcally mprovable procedure s vbratonal-rotatonal confguraton nteracton 21 (VRCI). The convergence of VRCI calculatons can be accelerated by optmzng the bass functons usng VSCF. 18 A partcularly effcent scheme, called vrtual confguraton nteracton (VCI) s to use a ground-state VSCF calculaton to generate sngle-mode functons and to use products of these sngle-mode functons (called modals) wth rotatonal bass functons as bass functons wth lnear coeffcents optmzed by the varatonal prncple. 20,21 These functons are called confguraton state functons (CSFs). In practce, as explaned below, we actually used ths procedure only for total angular momentum quantum number J equal to 0. For J > 0 the CSFs are constructed by takng the products of the J = 0 VCI egenvectors wth the rotatonal bass functons. A key advance 21 n systematzng of the procedure s to organze the calculaton usng a herarchcal representaton 20,25 of the potental and lmt the number of coupled modes n any ncluded term to two, three, or four. Carter and

3 3 Bowman 26 used VCI wth the herarchcal representaton to calculate about a hundred vbratonal levels for varous sotopologs of CH 4 for J = 0 and 9 levels for CH 4 for J = 1. Once the rovbratonal energy levels are obtaned by the VCI calculatons for all mportant values of the angular momentum J, they can also be used to compute partton functons ncludng full rotaton-vbraton couplng, but pror to the present paper ths has not been done. In a J = 0 calculaton, Bowman et. al 27 showed that ncluson of anharmonc terms sgnfcantly lowers the zero-pont energy of methane from ts harmonc oscllator zero-pont energy and ncreases the J = 0 partton functon by a factor of n the temperature range of K; they also estmated rotatonal effects by usng a calculaton of the lowest-energy J = 1 states to estmate the rotatonal constant for a separable-rotaton calculaton. 27 Manthe et. al 28 have also reported a J = 0 partton functon for methane that was obtaned usng a dfferent technque. The mportance of anharmoncty for the vbratonal energy of methane has also been shown n other recent work. 29,30 The present paper ncludes fully converged vbratonal states for J up to 50 n order to calculate a converged rotaton-vbraton partton functon over the temperature range K. Ths s the frst fully converged rotaton-vbraton partton functon for any molecule wth more than three atoms. The potental energy surface used for the present calculatons s that of Jordan and Glbert, 31 whch s based on older work by Raff 32 and Joseph et. al. 33 Although ths s not a quanttatvely accurate surface for methane, t s realstc enough for our purposes, and t has been used for recent rate constant calculatons on the hydrogen atom 27,34 50 and oxygen atom 51,52 abstractons of a hydrogen atom from methane. Our goal s to obtan accurate rotaton-vbraton partton functons for a gven realstc potental energy surface n order to assess the magntude of the rotaton-vbraton couplng, and the well studed Jordan-Glbert potental provdes an deal testng ground for ths purpose. Secton II summarzes the theoretcal formulaton used n the present work, and Secton III s devoted to the degeneracy and symmetry consderatons. The egenvalue calculatons were carred out usng a locally modfed verson of the MULTIMODE computer program, 53 and Secton IV provdes detals of these calculatons. Secton V contans detals of the partton functon calculatons. Secton VI presents the results and dscusson. The conclusons drawn from the present study are summarzed n Sec. VII.

4 4 II. QUANTUM MECHANICAL THEORY II.A. Applcaton of the VSCF method for J = 0 The complete Watson Hamltonan for a polyatomc molecule n normal coordnates s gven by 20 H Watson 2 F h h h = + ( J ) ( ) = 2 α πα µ αβ J β π β µ αα 2 k 1 Q 2 α= 1β= 1 8 α = 1 where (, β = x, y, z) α and Jα and k 2 3 ( Q, ) +, (1) V 1 K, Q F πα are the components of the total and vbratonal angular momentum operator respectvely, m s the effectve recprocal nerta tensor, Q k s the mass-weghted normal coordnate 54 for mode k, and F s the number of vbratonal degrees of freedom. The potental energy s a functon of the normal coordnates and s gven as V (, ) Q 1 K,Q F. The frst term n the above equaton s the knetc energy operator assocated wth each normal coordnate, the second term represents the couplng between the components of the angular momentum, and the thrd term, also known as the Watson term, s usually very small for polyatomc systems and s generally omtted from calculatons. (However, t s ncluded n the present work.) For a non-rotatng system, the VSCF method approxmates the vbratonal wave functon as a Hartree product of sngle-mode wave functons called modals ( Q ) Ψ F = 1 ( Q, K Q F ) = φ ( Q ) 1,, (2) where φ s the modal assocated wth normal coordnateq. The modals are constraned to be orthonormal: j j φ φ = δ, (3) whereδ j s the Kronecker delta. The VSCF method s a varatonal procedure for obtanng the modals, and the optmzed wave functon of the form n Eq. (2) s obtaned by mnmzng the total energy wth respect to all the modals subject to the constrant of Eq. (3), whch s enforced by the Lagrange multplers. Ths varatonal procedure gves a set of dfferental equatons for each modal

5 5 H SCF for =1,K,F, whereε s a Lagrange multpler. Because φ ( Q ) = ε φ ( Q ), (4) SCF H depends on the orbtals, ths s not a conventonal egenvalue problem; t s called a pseudo-egenvalue problem, and ε s the modal energy; these equatons are solved teratvely. Usng Eq. (1) wth total angular momentum J equal to zero, the SCF Hamltonan can be wrtten as the followng sum of knetc and potental energy operators SCF H T + U, (5) where T s the knetc energy operator assocated wth mode, and s gven as T and U s known as the mean feld operator and s gven as 2 2 = h, for =1,K,F, (6) 2 2 Q U = F 2 h 3 3 F φ j ( Q j ) V ( Q1, K, QF ) + πα µ αβ π β φ j ( Q j ). (7) j 2 α= 1β= 1 j In order to evaluateu, we have to perform an (F-1)-dmensonal ntegral over the normal coordnates, whch s computatonally ntensve for most polyatomc systems. To make the calculatons tractable, the potental energy term s expanded n a herarchcal fashon as 20 F F j (1) F ( 2) (3) 1,, QF ) = V ( Q ) + V, j ( Q, Q j ) + V, j, k ( Q, Q j, Qk ) = 1 = 1 j = 1 = 1 j = 1k = 1 V ( Q K F + V ( Q, Q, Q, Q ) + K (8) = 1j= 1k= 1l= 1 j k (4), j, k, l j k l By approxmatng the F-mode potental as a sum of one-mode, two-mode, three-mode, and four-mode terms, we have to evaluate only four-dmensonal ntegrals. In prncple, one should converge the expanson by ncludng hgher-order terms (fve-mode, sxmode,..), but experence 20,27 has shown that stoppng at three-mode couplng s sometmes already well converged. In the present artcle, we wll compare results obtaned wth three-mode couplng to those obtaned wth four-mode couplng.

6 6 The components of the vbratonal angular momentum operator depend on two normal coordnates va the Corols couplng constant and are expressed as F F α π α = ζ k,lqk, (9) Q l k=1 l=1 α whereζ k, l s the Corols couplng constant. 55 The treatment of ths term and the Watson term s explaned elsewhere. 25 II.B. Confguraton nteracton Snce the modes are coupled, one needs to go beyond the VSCF approxmaton. The egenfunctons of the ground-state SCF Hamltonan for J = 0 form an orthonormal bass, and the total vbratonal wave functon can be expanded n ths bass; as mentoned n the ntroducton, ths s called Vrtual CI (VCI). 20,21 For the calculatons n ths paper, we restrct the herarchcal expanson of Eq. (8) to at most four-mode couplng, and we form the VCI bass by one-mode, two-mode, three-mode, and four-mode exctatons from the ground state. The one-mode exctatons are lmted by specfyng the maxmum number of quanta each mode can possess. Two-mode, three-mode, and four-mode exctatons are lmted by two parameters; one of them s the maxmum number of quanta each mode can possess (called maxbas), and the other s the sum of quanta n all the modes (called maxsum). One could n prncple use symmetry to block dagonalze the Hamltonan, 26 but that was not done for the present calculatons. A general bass functon for the VCI calculaton s called a confguratonal state functon (CSF) and s wrtten as n 1 n2 K n KnF, where F s the number of modes (9 for methane) and n s the number of quanta n mode. All one-mode exctatons of the form K n 0, are ncluded, provded n maxbas(, 1). All two-mode excted K F states of the form Kn j K Kn 0 are ncluded, where the sum n + s less than F n j or equal to maxsum(2), and n and n j are less than or equal to maxbas(, 2) and maxbas(j, 2), respectvely. Smlarly, all three-mode and four-mode exctatons of the form K n Kn j Knk K0F and Kn K j nk Knl K0F s less than or equal to maxsum(3), and n K are ncluded, where n + n j + nk n n j + nk + nl + s less than or

7 7 equal to maxsum(4), respectvely, and also where n s less than or equal to maxbas(, 3) for three-mode exctatons and maxbas(, 4) for four-mode exctatons. 21,56 II.C. Applcaton of VCI method to J > 0 For the calculaton of rotatonal-vbratonal energy levels, the VCI scheme s appled wth the full Watson Hamltonan. The rovbratonal bass n whch the Watson Hamltonan s dagonalzed s obtaned by takng the drect product between the VCI bass functons and symmetrc-top wave functons. 20 The symmetrc-top wave functons are labeled by three quantum number J, K, M, where J s the angular momentum quantum number, K and M quantum numbers are assocated wth the projecton of the angular momentum along the body-fxed z-axs, and the space-fxed Z-axs, respectvely J 2 2 J, K, M = J ( J + 1) h J, K, M J z J, K, M = Kh J, K, M J Z J, K, M = M h J, K, M. (10) All exact egenvalues of the Watson Hamltonan are ndependent of M so we consder only M = 0, and we wrte J, K, 0 as J, K. Equaton (1) contans terms of the form J α J β where α, β = x, y, z, and the matrx elements of these operators n the J, K bass can be obtaned usng rasng and lowerng operators. The non-zero matrx elements of all combnatons of angular momentum operators occurrng n Watson Hamltonan have been gven earler by Bowman et. al. 20 and are shown n Appendx A. The matrx elements are non-zero only for K = 0, ± 1, ± 2. In the Watson Hamltonan, each of these terms also nvolve an element µ of the nverse moment of nerta tensor, and the αβ expressons n Eq. (1) that nvolve J x, J y, J z also nvolve the vbratonal angular momentum operators π α. After all the matrx elements of the Hamltonan are obtaned, the Hamltonan matrx s dagonalzed n ths rovbratonal bass and the rotatonvbraton energy levels are obtaned.

8 8 III. SYMMETRY AND DEGENERACY OF ROTATION-VIBRATION STATES Symmetry labelng of energy levels gves nformaton about the degeneraces assocated wth the energy levels. In the present calculatons, methane s treated as a molecule belongng to the C 1 pont group. Ths gves us an opportunty to numercally verfy the degeneraces assocated wth the vbratonal and rovbratonal energy levels of methane. Subsectons A and B present a dscusson on the symmetry of vbratonal and rovbratonal levels that s useful for analyzng the results. The ncluson of nuclear-spn degeneracy assocated wth rovbratonal levels plays an mportant role n the computaton of the partton functon and s dscussed Sec. III.C. III.A. Vbratonal symmetry Methane belongs to the T d pont group and has nne vbratonal degrees of freedom, whch have only four unque frequences. Of the nne vbratonal modes, there s one non-degenerate mode wth frequency υ 1, one doubly degenerate mode wth frequency υ 2, and two trply degenerate modes wth frequences υ3 and υ 4. Note that, n keepng wth the unversally accepted language, we sometmes use the word mode to refer to the nne component modes, but elsewhere (as n the rest of ths secton) t refers to the four (possbly degenerate) modes. Rather than ntroducng a new notaton when the above double usage s unversally accepted we smply cauton the reader about the context dependence of the word mode. The four modes wth unque frequences υ 1, υ2, υ3, and υ 4 can be labeled usng the rreducble representaton of the T d pont group, and the symmetry of the vbratonal wave functon can be obtaned by takng a drect product of these four symmetry labels. The sngle degenerate mode wth frequency υ 1 has symmetry A 1, the doubly degenerate mode wth frequency υ 2 has E symmetry, and each of the two trply degenerate modes have symmetry F 2. The symmetres of the overtone states of the non-degenerate modes are obtaned by takng a drect product of the symmetres of the fundamental states. When a mode s degenerate, the symmetres of ts overtone states are not obtaned smply by takng a drect product. 54 A detaled descrpton on ths topc s gven elsewhere 54 along wth a general expresson for obtanng the symmetry of overtone states of

9 9 degenerate modes for any pont group. The results of Herzberg 57a for the symmetry of overtone states of degenerate modes of methane are provded n the supportng nformaton. 58 The symmetry of a combnaton state s obtaned by smply takng the drect product of the ndvdual mode symmetres. For example, a combnaton state n whch there s one quantum each n υ 3 and υ 4 wll have a symmetry of A 1 + E + F 1 + F 2 and can be obtaned by takng a drect product of the F 2 rreducble representaton wth tself, but an overtone state wth two quanta n υ 3 and zero quanta n υ 4 wll span the A 1 + E + F 2 rreducble representatons. Fnally, f a combnaton state arses due to multple exctaton of both degenerate and non-degenerate modes, the symmetry can be obtaned by frst evaluatng the overtone symmetres of ndvdual modes usng the table n Ref. 57a or from the expresson n Ref. 54 and then by takng the drect product of the symmetres assocated wth the combnaton. Note that n the case of methane, evaluatng the drect product nvolvng the symmetry of υ 1 s of no consequence snce has A 1 symmetry. υ 1 III.B. Rovbratonal symmetry The rotatonal wave functon of any molecule can be labeled by the rreducble representatons of the D group, whch s the group of all rotatons and reflectons. The rreducble representatons g D J are used to represent all rotatonal states wth even J, and those of u D J are used to represent states wth odd values of J. To label the rotatonal u states of methane one has to reduce the representaton of and D J to the rreducble representatons of the Td pont group. 7 The overall symmetry of the rovbratonal wave functon s obtaned from the drect product of the symmetres assocated wth the vbratonal and rotatonal wave functons n the T d representaton. Under the harmoncoscllator rgd-rotator approxmaton, the degeneracy d assocated wth a generc rotaton-vbraton level of methane wth n quanta n each υ and a total angular momentum of J s gven as g D J d = (n 2 +1)(n 3 +1)(n 3 + 2)(n 4 +1)(n 4 + 2)(2J +1) 2 4. (11)

10 10 Here we have used the fact that for a harmonc oscllator, a state wth n quanta n a doubly degenerate mode s (n+1)-fold degenerate, and a trply degenerate state wth n quanta of exctaton s ((n+1)(n+2)/2)-fold degenerate, and the sphercal-top nature of methane gves the (2J + 1) 2 degeneracy assocated wth the rotatonal wave functon. The total degeneracy mentoned n Eq. (11) s preserved only for the dealzed case of a rgdrotator, harmonc oscllator Hamltonan. The presence of anharmonc effects and rotaton-vbraton nteractons lft some of the degeneracy. In addton, one must consder spn, as dscussed n Subsecton C. The effect of Corols couplng on the vbratonal levels of methane has been studed usng group theoretc methods n a seres of four papers by Jahn, 4 7 and only the results of those studes that are needed for the present work are summarzed here. It was shown by Jahn that the Corols couplng terms n the Watson Hamltonan transform accordng to the F 1 rreducble representaton of T d pont group. 6 As a consequence, two vbratonal states wll be coupled by Corols nteracton only when the drect product of ther rreducble representatons spans the F 1 rreducble representaton. 6,57b Usng the multplcaton table 54 for the T d pont group, the rreducble representatons spanned by A 1 µ A 1, E µ E, and F 2 µ F 2, are gven as A 1, A 1 + A 2 + E, and A 1 + E + F 1 + F 2, respectvely. Snce nether A 1 µ A 1 nor E µ E spans F 1, Corols splttng does not occur for non-degenerate and doubly degenerate modes of methane. It s only the two trply degenerate modes υ 3 and υ4 of F 2 symmetry n whch the three-fold degeneracy s lfted due to Corols couplng. The nteracton and the symmetry labelng of rovbratonal levels of υ 3 and υ 4 are best studed usng the rreducble representaton of the full rotaton-reflecton group; hence our frst task s to express the symmetry of trply degenerate modes usng the rreducble representatons of the full rotaton-reflecton group. It s shown n Ref. 7 and n the supportng nformaton 58 that u D 1 s the rreducble representaton for J = 1 n the D group, and spans F2 symmetry n the T d pont group. Snce the two trply degenerate modes υ 3 and υ 4 span F 2 symmetry n T d, one fnds that υ 3 and υ 4 span u D 1 n the D group. To obtan the symmetres of the rovbratonal levels, we have to obtan the

11 11 drect product between u D 1 and an rreducble representaton of a rotatonal state. One of the advantages of workng n the D group s the ease of evaluaton of drect products between two rreducble representatons. The drect product between two rreducble representatons of D was dscussed by both Wgner 59 and Hamermesh 60 and s summarzed by: u D 1 u D1 D D j = D J. (12) j 1 j1 + j2 J = j1 j2 g u g u DJ =D J 1 + DJ + DJ Usng the above equaton, the drect product between rotatonal states and the trply degenerate vbratonal states s gven as u g u g DJ = D J 1 + DJ + DJ + 1 (even J ), (odd J ). (13) The physcal nterpretaton of ths result s that the trply degenerate state has a vbratonal angular momentum assocated wth t and the vbratonal angular momentum nteracts wth the total angular momentum through the Corols couplng term; the vbratonal angular momentum can be parallel, perpendcular, or antparallel to the total angular momentum, whch splts the levels. The levels resultng from the splttng of the trply degenerate state are labeled as F, F, and F, respectvely. + 0 _ 57b III.C. Nuclear spn degeneracy The total wave functon must be ant-symmetrc wth respect to exchange of both the coordnates and spns of dentcal fermons, and we must take account of ths for the four dentcal protons n methane. As the Watson Hamltonan does not nclude any nuclear spn, the only effect of ncluson of nuclear spn functons wll be to ncrease the degeneracy assocated wth certan rovbratonal levels. A system of m dentcal partcles each wth a nuclear spn of I has a total of ( 2I + 1) m spn states, and therefore, for methane the total number of possble spn states s 16. Because the total wave functon must be ant-symmetrc wth respect to the exchange of any two hydrogen atoms n methane, not all of the 16-fold degeneracy s allowed for each rovbratonal state. In order

12 12 to fnd the correct nuclear spn degeneracy assocated wth each rovbratonal level, one has to evaluate a drect product between the permutaton group symmetres of the rovbratonal and nuclear spn states. The products that are totally symmetrc,.e., that belongs to the A 1 symmetry are the only combnatons that exst n nature. The symmetry of the nuclear spn functon for methane s 5A 1 + E + 3F 2, and Wlson has reported 61 a detaled descrpton of the statstcal weghts assocated wth the rovbratonal levels. However, for statstcal mechancal calculatons at temperatures at whch many rotatonal levels are occuped, one can replace the ndvdual weghts of the rovbratonal state by an average weght to all states. 61,62 The average weght s obtaned by dvdng the total nuclear spn multplcty (16 for methane) by the symmetry number (12 for methane). IV. EIGENVALUE CALCULATIONS All the calculatons were performed usng the potental energy surface of Jordan and Glbert, 31 whch was obtaned from the POTLIB database. 63,64 The rovbratonal energy levels were calculated by the VCI method summarzed n Sec. II; these calculatons were carred out usng a locally modfed verson of the MULTIMODE 53 program. The zero pont energy of the system was taken as the zero of energy for all tabulated energy levels. The frst step of the calculaton nvolves computaton of the J = 0 VSCF Hamltonan for the ground-state wave functon. In ths step the modals are expanded n harmonc oscllator functons; we used 12 harmonc oscllator functons n each mode. (A convergence check on ths value s presented n Appendx B) The egenvectors of the ground-state SCF Hamltonan were used to perform the VCI calculatons and the VCI matrx was constructed drectly from the VSCF modals. The number of bass functons used for ths purpose were controlled by nput parameters. As dscussed n Sec. II.B, the VCI bass s formed by usng a set of parameters called maxsum and maxbas. The maxmum sum of quanta for one-mode, twomode, three-mode, and four-mode couplng was fxed by gvng approprate values to maxsum(1), maxsum(2), maxsum(3), and maxsum(4). Then the maxmum allowed quanta n mode for one-mode, two-mode, three-mode, and four-mode exctatons was fxed by

13 13 settng maxbas(,1), maxbas(,2), maxbas(,3), and maxbas(,4) equal to maxsum(1), maxsum(1) 1, maxsum(1) - 2, and maxsum(1) - 3, respectvely. As an example, the procedure for obtanng the VCI bass for J = 0 s as follows. The maxmum sum of quanta was taken to be 7 for one-mode, two-mode, and three-mode exctatons, and 6 for four-mode exctatons. The maxmum allowed quanta n mode for one-mode, two-mode, three-mode, and four-mode exctaton was set to 7, 6, 5, and 4, respectvely. The resultng sze of the VCI matrx for J = 0 was In order to study the convergence we also performed calculatons wth smaller bases of szes 715, 868, 1372, 1876, 2065, 2905, 4165, and 4390, and the correspondng maxsum values are shown n Table I. For J > 0, the Hamltonan matrx was constructed by takng a drect product of the symmetrc-top rotatonal functons 20 wth the egenfunctons of the J = 0 VCI matrx. If the N Vb lowest-energy egenfunctons are chosen, the sze of the rovbratonal matrx for angular momentum J s gven by N (2J 1), and the rovbratonal energes are Vb + obtaned by dagonalzng a matrx of ths order. The values used for N Vb are specfed n Sec. VI. Once the rovbratonal bass s formed the matrx elements are computed usng the equatons n Appendx A. One can also use symmetry of the rovbratonal bass functon to expedte the process of formng the matrx, and a detaled descrpton s presented n Ref. 20. However for the present calculatons, methane was treated as a molecule of C 1 symmetry, and ths allowed the degeneraces n energy levels assocated wth T d pont group can be verfed numercally. V. PARTITION FUNCTION CALCULATIONS V.A. Accurate partton functon The canoncal partton functon was evaluated for a temperature range of 100 K to 1000 K by summng over all the rovbratonal states as (2I + 1) Q( T ) = σ m (2J + 1) J + J [ E ( v, J, K ) + E exp v kb K= J T where we have ntroduced a shorthand for the ground state energy E G E( 1 F = G ], (14) 0, K,0, J = 0, K 0), (15)

14 14 and I = 1,, 2 m = 4 σ s the symmetry number, the ndex v denotes the collecton of all the vbratonal quantum numbers, and E(v, J, K) s the M = 0 rovbratonal energy that was obtaned by the method dscussed n Sec. II (detals of the calculatons are gven n Sec. IV). The partton functon n Eq. (14) can be expressed as E G k T Q( T ) = Q( T ) e B. (16) We note for reference that the converged value of E G found n the present work s 9362 cm 1. In the above equaton, Q has the zero of energy at E G, and Q has the zero of energy at the mnmum valuev of the potental energy. The vbratonal partton functon e was calculated for the temperature range of K by summng over the computed vbratonal states, and the sum over J n Eq. (14) was carred out through J = 50. The test for convergence wth respect to J was done and the detals are provded n the supportng nformaton. 58 The partton functons calculated usng the rovbratonal levels obtaned by solvng the full Watson Hamltonan were labeled asq andq wthout subscrpts. V.B. Approxmatons to be tested Varous sets of approxmate partton functons were calculated usng the separable-rotaton approxmaton. Assumng separablty of rotatonal and vbratonal moton, the canoncal partton functon can be expressed as a product of vbratonal Q ) and rotatonal ( ) partton functons ( Vb Q Rot Q = Q Q, (17) SR where the subscrpt SR s used to ndcate that the partton functon s calculated usng the separable-rotaton approxmaton. The vbratonal partton functon was calculated usng two dfferent methods. In the frst method, Q was calculated from the harmonc frequences obtaned from normal mode analyss, and ths harmonc oscllator vbratonal partton functon was labeled as Q Vb, HO. The second method for obtanng QVb used the vbratonal energes obtaned by solvng the J = 0 Watson Hamltonan for the gven potental, and ths anharmonc approxmaton was labeled as Q Vb, 0. Vb Rot Vb J =

15 15 The rotatonal partton functon together wth the nuclear spn contrbuton for any nonlnear molecule s gven as m (2I + 1) + J Q Rot -Nuc = (2J + 1) exp[ ERot ( J, K) kbt ], (18) σ J K = J where J s the angular momentum quantum number, and K s the projecton of the angular momentum along a body-fxed z-axs. If we neglect centrfugal and Corols nteractons, the rotatonal energy of a sphercal top depends only on J: E Rot = BJ(J +1). (19) In Eq. (19), B s known as the spectroscopc rotatonal constant. Generally, B s evaluated from the prncpal moments of nerta at the equlbrum geometry, and then t s called B e. In the present work, the rotatonal partton functon was calculated usng two methods. In the frst case was used for calculatng the rotatonal energy by Eq. (19), and the nuclear-rotatonal partton functon obtaned from ths method was labeled as Q Rot-Nuc, e. In the second case, the Watson Hamltonan was solved for each J value, and the rotatonal energes were obtaned from the vbratonal ground-state energes at each J. There are ( 2 J +1) B e vbratonal ground state terms correspondng to K = J,K, + J, and these were substtuted n Eq. (18). The summaton was carred out for J = 1, K,50, and the computed nuclear-rotatonal partton functon was labeled asq. Rot-Nuc,0 By combnng the above treatments, four dfferent separable-rotaton partton functons were obtaned and are summarzed as follows: (W,G) = Q Q, (20) QSR Vb, J = 0 = QSR (W, Be ) QVb, J = 0 SR Vb, HO Rot -Nuc,0 Q Rot -Nuc,e Rot -Nuc,0, (21) Q (HO,G) = Q Q, (22) Q (HO, B ) = Q Q, (23) SR e Vb, HO Rot-Nuc,e where W denotes the use of the Watson Hamltonan for J = 0, and G denotes the use of the ground vbratonal state for each J.

16 16 For reference we note that the harmonc approxmaton to E G yelds 9530 cm 1 for the present potental energy surface. Usng ths value and Eqs. (16) and (20)-(23), we can also obtan four approxmatons to Q (T ), namely ( W, G), ( W, B e), ( HO, G ), and ( HO, Be ). VI. RESULTS AND DISCUSSION The zero pont energy and the fundamental exctaton energes for each of the four modes are shown n Table II. The average energes and the standard devaton ( ) of some excted vbratonal levels are shown n Table III. Note that under the harmonc oscllator approxmaton, each vbratonal state dscussed n Table III wll be d-fold degenerate. (From ths pont on, we are dscussng only M = 0 states. The full degeneracy s always ( 2 J +1) tmes greater due to M degeneracy.) The value of d s 0 (2J +1) where d d0 can be obtaned by solvng Eq. (11) wth J = 0 for each vbratonal level. However, the presence of anharmonc terms couples the normal modes resultng n partal loss of the d-fold degeneracy. A detaled descrpton of the nfluence of anharmoncty on degenerate vbratonal states of Td molecules s gven n Ref. 57c. It should be noted that lftng of the d-fold degeneracy s partal, and some states do not lose ther degeneracy due to anharmoncty. Table III llustrates ths for several vbratonal states whose degeneracy s splt by anharmoncty. The standard devaton for each group of states consdered n Table III was computed usng the followng expresson = d 2 ( E E ) d 1, (24) where E s the energy of the state, and E s the average energy, and d s the degeneracy n the absence of anharmoncty. Table III also shows the trend n the average energes and standard devaton wth respect to the change n the VCI bass sze. It s seen that ncreasng the bass has very lttle effect on the values. It has been found that the average energy of states does not decrease monotoncally wth the VCI bass sze. Ths s because on ncreasng the VCI bass sze new energy levels are ntroduced whch were mssng n the smaller bass. Incorporaton of new states changes the densty of states assocated wth a gven energy level. The densty of states assocated wth the average

17 17 energes of 5650 VCI bass sze s dscussed n the supportng nformaton. 58 It was also found that the densty of states ncreases wth ncreasng energy. As dscussed n Sec. III.B, no Corols splttng occurs for vbratonal states of A 1 and E symmetres, and hence these states are expected to be ( 2J + 1) and 2 (2J + 1) fold degenerate, respectvely, for any J,.e., d0 s 1 for A1 states and 2 for E states. In Table IV, the values for the average energy of the vbratonal ground state, sngly excted υ2 (E) state, and sngly-excted υ 4 ( F 2 ) state are lsted for a few selected J values along wth ther respectve standard devaton. The averages and the standard devaton at each J value for the states were computed over ( 2J + 1), 2 (2J + 1), and 3( 2 J +1) values, respectvely. The A1 and E states showed a much smaller devaton from ther respectve mean values as compared to the F 2 state, ndcatng that three-fold degeneracy of the F 2 state was removed by Corols couplng. Snce both A 1 and E states are strctly degenerate states, they should have zero standard devaton, but the results shown n Table IV have non-zero standard devaton due to the numercal methods used for computng them. Ths ssue s dscussed for the recent paper on H 3 O + and D 3 O The effect of rotaton-vbraton couplng was also studed for varous J values; the detals of these studes are presented n both Appendx B and supportng nformaton. 58 As dscussed n Sec III.B, group theoretcal methods provde us wth the degeneraces assocated wth varous rovbratonal levels, and the numercal results were found to be n good agreement wth the values predcted usng group theory. The effect of usng a 3- mode and a 4-mode representaton on rovbratonal energes was studed usng J = 20 as an example. It was found that by ncreasng the representaton from 3-mode to 4-mode, the mnmum and the maxmum rovbratonal energes decreased by 2.5 and 5.7 cm -1, respectvely. Ths change corresponds to about 0.1% and s consdered to be very small. The expresson for the partton functon can be rewrtten as Q T = Q ( ) J ( T ), (25) where Q J s the contrbuton from each J level and s defned as J Q J (2I + 1) ( T ) = (2J + 1) σ m + J E( v, J, K) exp. (26) v kb K= J T

18 18 Snce the computatonal effort to solve for the egenvalues of the Watson Hamltonan wth a gven J ncreases as J 2, a compromse was acheved between computatonal effort and bass-set optmzaton effort by dvdng the range of angular momentum under study nto subsets and usng a dfferent value of N Vb for each of them. The sze of the rovbratonal matrx s of the order of N Vb (2J +1), and the dagonalzaton of the rovbratonal matrx becomes computatonally expensve for a hgh value of J and N Vb. In order to keep the calculaton tractable a converged value of N Vb was obtaned for the hghest J value for each subset. For example, a converged value of N Vb was obtaned for J = 10 and was used for the set of J values n the range 5 < J < 10. A smlar procedure was used for J = 15, 20, and 25, and converged values for Q were obtaned. A lst of N Vb values used at dfferent J states s provded n Table V, and the detals of the convergence studes for these J states are summarzed n Table VI, VII, and Appendx C. The convergence studes were carred out for the temperature range of K and converged values Q J were obtaned wth respect to ncreasng the sze of both the VCI and the rovbratonal bass. For example, Table VI compares the Q J values for J = 10 obtaned from varous VCI bases. It s seen that a VCI bass sze of 5650 gves converged values over the entre temperature range. In these calculatons, the rovbratonal matrx was formed by usng only the lowest The N Vb J functons out of the full 5650 VCI functons. N value s lsted n Table V, and for J = 10, was taken to be 500. The sze Vb of the rovbratonal matrx formed s of order N Vb N ( 2 J 1), and for J = 10 a Vb + rovbratonal matrx of sze was dagonalzed, and the rovbratonal energes so obtaned were used for calculatng Q J. Convergence wth respect to N Vb was checked by formng the rovbratonal matrx usng half the number of VCI functons. For J = 10, the value of N was reduced from 500 to 250 functons and the resultng Vb sze of the rovbratonal matrx of was obtaned. As seen from Table VII, a rovbratonal matrx that has only one quarter as many elements leads to a Q J value that dffer by less then 0.01% from the one computed wth the larger bass. All the energy

19 19 5 levels that contrbuted more than or equal to 10 at 1000 K were nclude n the summaton n Eq. (26). Ths corresponds to ncluson n the summaton of all states that are below 8000 cm -1. Because of the mportance of E G for practcal calculatons, tests of ts convergence are provded n the Appendx B, along wth tests of the convergence of selected ndvdual vbratonal energy levels for J = The computed vbratonal partton functons are shown n Table VIII. A converged value of Q Vb, J = 0 for 1000 K was obtaned for a VCI matrx sze of The vbratonal partton functon was computed usng both 3-mode and 4-mode representatons of the potental energy, and the results are gven n Table IX; the 3-mode representaton was found to be suffcently accurate. In partcular the dfferences of the 3- mode and the 4-mode results were less that 1%. Convergence wth respect to the number of harmonc oscllator functons was verfed, and vbratonal partton functons obtaned usng 12 and 6 harmonc oscllator functons per modal were found to be wthn 0.1% of each other. A smlar test was also performed for the Gauss-Hermte ntegraton ponts, and the vbratonal partton functon obtaned usng 30 and 15 ponts were wthn 0.1% of each other. The computed rovbratonal partton functon and the separable-rotaton partton functon are summarzed n Table X. It was found that n the low-temperature regon, the rgd-rotator harmonc oscllator Q SR ( HO, Be ) partton functon s very close to the accurate rovbratonal partton functon (Q), but the HO, B approxmaton begns to show sgnfcant errors as the temperature s ncreased, reachng 2% at 400 K and 9% at 1000 K. The separable-rotaton partton functon Q (W,G), as descrbed n Sec. IV, was found to agree closely wth the accurate rotatonal-vbratonal Q for all temperatures; thus ths s an nexpensve alternatve for computaton of accurate rovbratonal partton functon. Because the method requres only the vbratonal ground state energes for each of the J > 0 values, t does not requre a large VCI and rovbratonal bass. SR e

20 20 The zero-pont nclusve partton functon s lsted n Table XI, and t was found that the accurate rovbratonal partton functon s greater than the rgd-rotator harmonc oscllator partton functon by a factor of 11.4 and 1.4 at 100 K and 1000 K, respectvely. At 300 K the popular harmonc oscllator, rgd-rotator approxmaton underestmates the partton functon Q by a factor of 2.3. Although Q s a more nterestng quantty from the pont of vew of statstcal mechancs (and s the quantty appearng n the textbooks), Q s the more nterestng quantty from the pont of vew of practcal applcatons because errors n calculatng the zero pont energy are equally as problematc as errors n calculatng the thermal contrbutons. Agan, the W,G approxmaton performs well. We should keep n mnd, though, that CH4 s probably close to a best case scenaro for separable-rotaton approxmatons n that the lack of any low-frequency modes greatly decreases the mportance of rotaton-vbraton couplng. Now that we have demonstrated the feasblty of full thermodynamc rotaton-vbraton calculatons for a pentatomc molecule, t wll be nterestng to test approxmate theores for molecules wth lower frequences and large-ampltude moton. VII. CONCLUSIONS Fully converged rotatonal-vbratonal partton functons of methane were computed by summng over the rovbratonal levels for the temperature range of K, and the accurate results were compared wth partton functons obtaned usng the separable-rotaton approxmaton. The egenvalues of the full Watson Hamltonan were obtaned usng the computer program MULTIMODE and were converged wth respect to VCI bass. The egenvalues also showed the expected trends n degeneracy for a gven J value. The dfference n vbratonal partton functon for 3-mode and 4-mode expanson of the potental was found to be neglgble for the present work. VIII. ACKNOWLEDGMENTS Ths work was supported n part by the Natonal Scence Foundaton under grant no. CHE , and by the offce of Naval Research (ONR-N ).

21 21 APPENDIX A The angular momentum matrx elements needed for the VCI calculatons are as follows: 20 JK Jz JK = K (A1) 2 2 JK J z JK = K (A2) JK J 2 x JK = JK J 2 y JK 1 2 = [ J ( J + 1) K ] (A3) 2 JK ±1J x JK = m JK±1J y JK 1 2 = m [( J m K)( J ± K + 1)] (A4) 2 2 x JK ± 2 J JK = JK ± 2 J 2 y JK = [( J ± K + 1)( J ± K + 2)( J m K)( J m K 1)] (A5) 4 JK J x J y JK = JK J y J x JK = (K 2) (A6) JK ± 1J z J x JK = m JK ± 1J z J y JK = (K ±1) JK ±1J x JK (A7) JK ± 1J x J z JK = m JK ± 1J y J z JK = K JK ±1J JK (A8) JK ± 2 J x J y JK = JK ± 2 J y J x JK 1 2 = m [( J m K 1)( J ± K + 2)( J m K)( J ± K + 1)] (A9) 4 Notce that we have corrected two typos n Ref. 20. x

22 22 APPENDIX B Table B-I provdes the detals of the convergence rate of the ground-state energy E G usng dfferent VCI bases. The table also compares the E G values obtaned usng 3-mode and 4-mode representatons of the potental energy term. The ground-state energy obtaned usng dfferent harmonc oscllator functons for each mode s also lsted. Comparsons of the convergence for levels wth J > 0 are more cumbersome because of the splttng assocated wth the (2J + 1) values of the energy correspondng to dfferent values of K for a gven set of vbratonal quantum numbers and a gven total angular momentum J; however t s nterestng to compare the convergence of the lowest-energy and hghest-energy K state, and ths s done n Table B-II for J = 10. Smlar comparson for hgher values of J = 15, 20, and 25 s provded n the supportng nformaton. 58 Table B-III and B-IV lst the rovbratonal states assocated wth the 0001 vbratonal state for J =1 and 20, respectvely. The 0001 vbratonal state s three-fold degenerate and belongs to the F 2 rreducble representaton of the T d pont group. (As n man text, we dscuss only M = 0 states; there s an addtonal degeneracy of a factor ( 2 J +1) due to M states, but ths factor s not ncluded n the present dscusson.) The number rovbratonal state assocated wth the 0001 vbratonal state for any value of J s gven by3 ( 2J +1). Usng ths relaton, the total number of rovbratonal states for J = 1 and 20 are 9 and 123, respectvely. As dscussed n Sec. III.B and Eq. (13), the rovbratonal states can be labeled usng the rreducble representatons of the D symmetry group. The F 2 state of T d pont group transforms accordng to the u D 1 rreducble representaton of the D group. The rovbratonal states assocated wth any J value are gven as

23 23 u g u g u D1 D J = D J 1 + D J + D J + 1 (even J ), (B1) u u g u g D1 DJ = DJ 1 + DJ + DJ + 1(odd J ). (B2) For J = 1 and 20, the rovbratonal states are gvens as, u u g u g D 1 D1 = D0 + D1 + D2, (B3) u g u g u D 1 D20 = D19 + D20 + D21. (B4) The rovbratonal levels assocated wth each rreducble representaton are gven by ( 2J + 1) g D 0 u D 1 states. For J = 1 case, the,, and representatons contrbute 1, 3, and 5 states, u D 19 g D 20 g D 2 u D 21 respectvely. For J = 20, the,, and representatons contrbute 39, 41, and 43 states, respectvely, towards the total of 123 sates. The representatons can then be expressed n terms of the rreducble representaton of the Td pont usng the followng relatons 61 g D0 = A1 D u 1 = F2 g D2 = E + F2, (B5) (B6) (B7) u D19 = 3 A1 + 3E + 10F2 g D20 = 3 A1 + 4E + 10F2 u D21 = 4 A1 + 3E + 11F2 (B8) (B9). (B10) The rovbratonal levels assocated wth the 0001 vbratonal state wth J = 1 are presented n Table B-III. The rovbratonal states are symmetry labeled as A 1 + E + 2F 2 and exhbt the degeneraces assocated wth each symmetry label. For J = 20, the number of

24 24 u D 19 g D 20 states arsng from the,, and representatons are 39, 41, and 43, respectvely, and are lsted n Tables B-IV. It was found that the dfference n energes of any two states belongng to dfferent rreducble representaton was very hgh as compared to the energy dfference between consecutve states belongng to same rreducble representatons. Because of the small energy dfference between consecutve states t was not possble to assgn T u D 21 u D 21 symmetry labels to each of them, but the apprecable energy dfference between the states u D 19 g D 20 belongng to,, and allowed us to dvde the 123 rovbratonal states n groups d of 39, 41 and 43 states as predcted from the group theoretcal treatment. The rovbratonal energes obtaned usng the 4-mode representatons are also shown n Table B-IV, and t was found that for the gven set of rovbratonal energes the 3-mode and the 4-mode representatons gave converged results. APPENDIX C Ths appendx shows convergence studes smlar to Table VI and VII (dscussed n Sec. VI) but for J = 15, 20, and 25. These results are n Tables C-I to C-VI.

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28 28 Table I. Values of maxsum used for formng varous szes of VCI bass for J = 0. maxsum(1,2,3,4) VCI sze

29 29 Table II. Fundamental exctaton energes for J = 0. a,b υ 1 υ 2 υ 3 υ 4 Energy (cm -1 ) 1000 (A 1 ) (E) (F 2 ) (F 2 ) a Calculatons were performed usng 3-mode representaton wth 15 Gauss-Hermte ntegraton ponts and 12 harmonc oscllator functons per mode. b The zero pont energy s 9362 cm -1.

30 30 Table III. Average energy (n cm -1 ) and standard devaton a of excted vbratonal states for J = 0 wth dfferent szes of VCI bass. b υ 1 υ 2 υ 3 υ 4 d c a Calculated usng Eq. (24). b Calculatons were performed usng 3-mode representaton wth 15 Gauss-Hermte ntegraton ponts and 12 harmonc oscllator functons per mode. c Ths s the degeneracy that the level would have n the absence of anharmoncty; t s the value used n Eq. (24).