THE JOURNAL OF CHEMICAL PHYSICS 139, 164315 (2013)

Analytic Morse/long-range potential energy surfaces and predictedinfrared spectra for CO–H2 dimer and frequency shifts of COin (para-H2)N N = 1–20 clusters

Hui Li,1,2,a) Xiao-Long Zhang,1 Robert J. Le Roy,2,b) and Pierre-Nicholas Roy2,c)

1Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry,Jilin University, 2519 Jiefang Road, Changchun 130023, People’s Republic of China2Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

(Received 22 August 2013; accepted 10 October 2013; published online 30 October 2013)

A five-dimensional ab initio potential energy surface (PES) for CO–H2 that explicitly incorpo-rates dependence on the stretch coordinate of the CO monomer has been calculated. Analytic four-dimensional PESs are obtained by least-squares fitting vibrationally averaged interaction energiesfor vCO =0 and 1 to the Morse/long-range potential function form. These fits to 30 206 points haveroot-mean-square (RMS) deviations of 0.087 and 0.082 cm−1, and require only 196 parameters. Theresulting vibrationally averaged PESs provide good representations of the experimental infrared data:for infrared transitions of para H2–CO and ortho H2–CO, the RMS discrepancies are only 0.007 and0.023 cm−1, which are almost in the same accuracy as those values of 0.010 and 0.018 cm−1 obtainedfrom full six-dimensional ab initio PESs of V12 [P. Jankowski, A. R. W. McKellar, and K. Szalewicz,Science 336, 1147 (2012)]. The calculated infrared band origin shift associated with the fundamentalof CO is −0.179 cm−1 for para H2–CO, which is the same value as that extrapolated experimentalvalue, and slightly better than the value of −0.176 cm−1 obtained from V12 PESs. With these po-tentials, the path integral Monte Carlo algorithm and a first order perturbation theory estimate areused to simulate the CO vibrational band origin frequency shifts of CO in (para H2)N–CO clustersfor N =1–20. The predicted vibrational frequency shifts are in excellent agreement with availableexperimental observations. Comparisons are also made between these model potentials. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4826595]

I. INTRODUCTION

Spectroscopic studies on cold helium clusters doped witha single chromophore molecule such as CO2, N2O, and OCSas the member of the “carbon dioxide family”, have been usedto probe microscopic superfluidity.1–21 In similar way, clus-ters of para-hydrogen (para H2) molecules doped with a sin-gle chromophore molecule22–33 are also considered as a pos-sible route to investigate the superfluidity of para-hydrogen.Recently, combining experimental measurements and theo-retical simulations, the non-classical rotational inertia and su-perfluid response in pure para H2 clusters doped with CO2

have been first elucidated.29 However, the size-dependent su-perfluid responses of these clusters reached a maximum atN = 12 para-H2 particles, and the clusters become frozen atlarger N due to localization caused by relatively strong inter-actions between CO2 and para H2. Carbon monoxide is a gen-tler probe molecule with much weaker and less anisotropicinteraction with para H2, and a relatively large rotational con-stant. These features lead to a delocalized distribution ofpara H2 molecules with respect to CO.34 An accurate descrip-tion of binary complexes is an essential starting point for theexploration of larger clusters. Indeed, the results of quantumMonte Carlo simulations of doped He clusters are known to

a)E-mail: prof_huili@jlu.edu.cnb)E-mail: leroy@uwaterloo.cac)E-mail: pnroy@uwaterloo.ca

be very sensitive to the quality of the pair potentials utilizedfor the calculations.35, 36

The CO–H2 complex, due to its astrophysicalimportance37 and the fact that it is an interesting test casefor the empirical and ab initio determination of intermolec-ular forces,38–44 has also been the subject of considerableexperimental and theoretical attention.45–51 The efforts havebeen summarized in an earlier paper by Jankowski andSzalewicz.43 The first infrared spectrum of para H2–CO andortho D2–CO complexes in the region of the fundamentalband of CO was recorded and assigned by McKellar in1998.50, 51 However, after more than ten years, the recordedinfrared spectrum of ortho H2–CO (or para D2–CO) had notbeen assigned yet due to its more complicated structure thanthat of para H2–CO system. Thus, a theoretical prediction ofthe infrared spectrum of ortho H2–CO (or para D2–CO) isvery necessary to help assign the recorded experimental data.Reliable theoretical predictions depend on the availabilityof an accurate potential energy surface for the CO–H2

complex. Two recent theoretical potential energy surfacesfor this complex have been reported.43, 44 One is based on afour-dimensional (4D) potential energy surface (PES) thatuses symmetry-adapted perturbation theory (SAPT), withCO fixed at its equilibrium geometry, referred to hereafterV98;43 however, although a 4D treatment may be adequatefor describing the microwave spectrum of ground-statespecies, it cannot properly describe infrared spectra involving

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excitation of an intra-molecular C–O vibrational mode.The other PES is a 5D ab initio potential which explicitlyaccounts for the H–H stretch vibrational motion while theC–O stretch coordinate is fixed at its ground vibrational stateaveraged distance, referred to hereafter V04;44 however, theexperimentally recorded infrared spectra were in the region ofthe fundamental band of C–O. Very recently, a new ab initiointeraction potential for the CO–H2 complex computedon a six-dimensional ab initio grid was presented.52, 53

The predicted infrared spectra calculated from this surfacehave already been shown to agree extremely well with theexperimental spectra of the para and ortho H2–CO complex,and enabled an assignment of the experimental spectrum ofortho H2–CO complex, which had been measured more thanten years ago. Because the experimentally recorded infraredspectrum only involved the excitation of the intra-molecularC–O vibrational mode, building an effective five-dimensional(5D) potential energy surface, which explicitly takes accountof the C–O stretch coordinate is necessary, and it may beadequate to properly describe the infrared spectrum. In thispaper, we endeavour to build a 5D ab initio surface whichonly explicitly takes account of the C–O stretch coordinatewith H–H stretch fixed at its averaged distance in the groundstate. In order to assess the accuracy of the interaction, wedetermine the microwave and infrared spectral transitions onthis surface for the para and ortho H2–CO complexes, andcompare to experimental and existing theoretical results.

A few years ago, the vibration-rotation transitions of(para-H2)N–CO have been studied,25 while the most secureexperimental assignments for those clusters are only limitedto N = 1–6. The vibrational band origin shift when goingfrom N = 0 to 1 is −0.18 cm−1, thus they simply assumedthat the band origin shifts for larger clusters are also linearwith N × 0.18 cm−1. With this rough approximation, their as-signments extended to the first solvation shell at about N = 15.This also stimulates us to generate a new, highly accuratestate-of-the-art ab initio surface for the H2–CO complex,which explicitly incorporates the stretch vibrational motion ofC–O and could be used to simulate the band origin shifts forlarger clusters. In this paper, the vibrational frequency shiftsof CO in para H2 clusters as a function of N are predicted onour 5D and recent full 6D V12 potentials, respectively, and acomparison between the two theoretical simulations and ex-perimental observations is performed.

Recently, Le Roy et al. introduced the “Morse/long-range” (MLR) radial potential function form which incorpo-rates theoretically known long-range inverse-power behaviourwithin a single smooth and flexible analytic function.54, 55

For atom–molecule or molecule–molecule systems, allow-ing parameters of that radial function to vary with angleand monomer-stretching coordinate yields a compact andflexible multi-dimensional functional form. Application ofthis approach to the CO2–He and CO2–H2 systems yieldeda function that explicitly incorporates the Q3 asymmetric-stretch vibrational motion of CO2, and has the correct angle-dependent inverse-power long-range behaviour.35, 56–58 Vibra-tionally averaging over Q3 for different vibrational levelsof the CO2 monomer yielded analogous 2D or 4D formsand led to remarkably accurate predictions of the vibra-

tional frequency shifts of CO2 in (He)N35 or (para H2)N,29, 30

respectively.In the present work, 4D versions of “Morse/long-range”

(MLR) functions (depending on three angles and R) have beenfitted to vibrationally averaged interaction energies obtainedfrom new five-dimensional ab initio PESs for CO–H2 whichexplicitly incorporate the stretch vibrational motion of C–O.The new ab initio calculations and the techniques used forcomputing the eigenvalues of the resulting potential energysurface are described in Sec. II. Section III then presentsour analytic four-dimensional potential function form and de-scribes its fit to the ab initio results. Section IV presents pre-dictions of the infrared and microwave spectra for the CO–H2

bimer implied by this surface, and compares them with exper-iment. Calculated vibrational frequency shifts for the doped(para H2)N–CO are also presented in Sec. IV. Concluding re-marks are given in Sec. V.

II. COMPUTATIONAL METHODS

A. ab initio calculations

The geometry of a CO–H2 complex in which CO isrigidly linear can be described naturally using the Jacobi coor-dinates (R, θ1, θ2, φ, rCO) shown in Fig. 1; there, �R is a vec-tor pointing from the centre of mass of CO to the centre ofmass of H2, θ1 the angle between �R and a vector pointingfrom atom O to atom C, θ2 the angle between �R and a vec-tor pointing from H(2) atom to H(1), φ the dihedral angle be-tween the two planes defined by �R with the CO molecule andwith the H2 molecule, rH2 the bond length of the H2 molecule,which was fixed at the average value for the ground state, 〈 r 〉= 0.7666393 Å,59 and rCO is the coordinate for stretch vibra-tion of CO.

In a full five-dimensional treatment which also took ac-count of the intra-molecular stretch coordinate rCO, the totalpotential energy for CO–H2 would be written as

V (R, θ1, θ2, φ, rCO) = VCO(rCO) + �V (R, θ1, θ2, φ, rCO)(1)

in which VCO(rCO) is the 1-dimensional (1D) potential en-ergy for stretching of an isolated linear CO molecule, and�V (R, θ1, θ2, φ, rCO) is the intermolecular interaction po-tential. The 1D potentials VCO(rCO) governing the vibration ofthe CO monomer were generated by least-squares fits to the

CO

H(2) H(1)

θ

θ

φ

R

rCO

rH2 2

1

→

FIG. 1. Jacobi coordinates for CO–H2 complex.

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164315-3 Li et al. J. Chem. Phys. 139, 164315 (2013)

experimental spectra data.60 The calculated fundamental tran-sition on this potential is 2143.3387 cm−1, which results ingood agreement with experimental value of 2143.2712 cm−1.The intermolecular potential energies of CO–H2 were cal-culated using single- and double-excitation coupled-clustertheory with a non-iterative perturbation treatment of tripleexcitations[CCSD(T)].61 The basis set used was the aug-mented correlation-consistent polarized n-zeta basis set ofWoon and Dunning (denoted as aug-cc-pVnZ or AVnZ forn = 3 and 4),62 supplemented with an additional set of bondfunctions (3s3p2d1f1g) (where α = 0.9, 0.3, 0.1 for 3s and3p; α = 0.6, 0.2 for 2d; α = 0.3 for f and g) placed at themidpoint of the intermolecular axis R.63, 64 The supermoleculeapproach was used to produce the intermolecular potentialenergies �V (R, θ1, θ2, φ, rCO), which is defined as the dif-ference between the energy of the CO–H2 complex and thesum of the energies of the CO and H2 monomers. The fullcounterpoise procedure was employed to correct for basis setsuperposition error (BSSE).65

The total intermolecular interaction potential �Vint canbe expressed as

�Vint = �V HFint + �V corr

int (2)

in which,

�V corrint = �V

CCSD(T)int + �V

T(Q)int . (3)

The Hartree-Fock part �V HFint is extrapolated by using two-

point formula of (n + 1) · exp(−9√

n),66 where n is the socalled cardinal number of aug-cc-pVnZ basis set with n = 3and 4. The correlation energy �V

CCSD(T)int is obtained directly

from the two-point 1/n3 extrapolation. Concerning the higherorder effects [beyond CCSD(T)], van Heusden et al. alreadynoticed that molecules with triple bonds are particularly sen-sitive to electronic correlation in high orders of perturbationtheory.67

Recently, Noga et al. examined the importance of higherorder contributions to the interaction potential of CO–H2 atmost important minima of the interaction energy surface. Theresults indicate that both the missing contributions from thetriple excitations and from the quadruple excitations affectthe final potential appreciably and anisotropically. Their to-tal value in the vicinity of the global minimum amounts toabout 3 cm−1, whereas only about 1 cm−1 at the secondaryminimum.68 In this work, electron correlation energies fromthe triple and quadruple excitations �V

T(Q)int are calculated

at CCSDT(Q) level using aug-cc-pVDZ basis set withoutbond function. Convergence study have been performed inRef. 53. All calculations were carried out using the MOLPROpackage,69 and MRCC program of Kallay and Surjn.70

The calculations were performed on regular grids for allfive degrees of freedom. Five grid points corresponding to rCO

= 1.052410, 1.101292, 1.147176, 1.194799, and 1.249673 Åwere chosen for the CO stretching coordinate, while a rela-tively dense grid of 29 points ranging from 2.4 to 10.0 Å wasused for the R intermolecular coordinate. The angular coordi-nates θ1 and θ2 range from 0 to 180◦ with step sizes of 15◦,and the dihedral angle φ ranges from 0 to 90◦ at intervals of30◦. This gives a total of 171 535 ab initio points to yield a5D PESs. The CCSDT(Q) calculations scale with the number

of basis functions N as N9, which is two order of magnitudegreater than the scale of CCSD(T) calculations with N7, andhence full five-dimension calculations at CCSDT(Q) level be-come soon computationally not executable even for small sys-tems with basis sets. In this work, correlation energies �V

T(Q)int

are only calculated on 4D intermolecular coordinates withC–O stretch fixed at its averaged distance of vibrationalground state. A relatively sparse grid point of 10 points rangefrom 2.4 to 10.0 Å was used for the R. The angular coordi-nates θ1 range from 0 to 180◦ with step sizes of 15◦, while θ2

range from 0 to 180◦ with step sizes of 30◦ and the dihedralangle φ ranges from 0 to 90◦ at intervals of 30◦. This gives atotal of 3694 ab initio points to yield 4D PESs for �V

T(Q)int .

B. Hamiltonian and reduced-dimension treatment

Within the Born-Oppenheimer approximation, withoutseparating the intra- and intermolecular vibrations, the ro-vibrational Hamiltonian of the CO–H2 complex in the space-fixed frame has the form (in a.u.):57, 71–73

H = − 1

2μ

∂2

∂R2− 1

2M

∂2

∂r2CO

+ l21

2ICO+ BH2 l

22

+ (J − l1 − l2)2

2μR2+ V (R, θ1, θ2, φ, rCO) (4)

in which μ−1 = (2mH)−1 + (mO + mC)−1 and M = mC mO/(mO + mC), where mH, mC, and mO are the masses of the H,C, and O atoms,74 respectively, BH2 is the inertia rotationalconstant of H2, ICO is the moment of inertia of an isolatedCO molecule, and V (R, θ1, θ2, φ, rCO) is the total potentialenergy of the system.

The above Hamiltonian incorporates full coupling be-tween the intermolecular and rCO vibrations. However, con-vergence of the eigenvalue calculations is very slow at thehigh internal energies associated with excitation of the vibra-tion of CO, since it requires a relatively large number of Lanc-zos iterations.56, 75 It is therefore highly desirable to separatethe treatment of the inter- and intramolecular motions. Sincethe vibrational mode of CO has a much higher frequency thando the intermolecular modes, Born-Oppenheimer separationtype arguments suggest that it should be a good approxima-tion to introduce such a separation, as long as the off-diagonalvibrational coupling is sufficiently small.56, 75 In this approx-imation, the total vibrational wave function would be writtenas the product

�v(R, θ1, θ2, φ, rCO) = φv(R, θ1, θ2, φ)ψv(rCO) (5)

in which v is quantum number for a specific stretching vibra-tional state of the free CO molecule, and the associated 1Dvibrational wavefunction ψv(rCO) is obtained by solving the1D Schrödinger equation:[ −1

2M

d2

dr2CO

+ VCO(rCO)

]ψv(rCO) = Ev ψv(rCO). (6)

The present work focuses on complexes formed fromCO in the ground (v = 0) and first excited (v = 1) stretch-ing states of CO. Using Eq. (5), the vibrationally averaged

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CO–H2 interaction potential for CO in vibrational level v is

V [v](R, θ1, θ2, φ)

=∫ ∞

−∞ψ∗

v (rCO)�V (R, θ1, θ2, φ, rCO)ψv(rCO)drCO (7)

and the associated four-dimensional intermolecular Hamilto-nian in the space-fixed reference frame is

H = − 1

2μ

∂2

∂R2+ Bv

CO l21 + BH2 l

22

+(J − l1 − l2

)2

2μR2+ V [v](R, θ1, θ2, φ) (8)

in which

BvCO = 〈ψv| 1

2 I (rCO)|ψv〉 (9)

is the CO inertial rotational constant and I(rCO) is the instan-taneous CO moment of inertia. Note that the vibrationally av-eraged intermolecular potentials V [v](R, θ1, θ2, φ) for differ-ent values of v differ both because the wavefunctions ψv(rCO)are associated with different values of v.

In order to solve our 4D Schrödinger equation numeri-cally in terms of the body-fixed angles (θ1, θ2, φ), the Hamil-tonian in the body-fixed reference frame is written as13, 76–78

H = Tstr + Tdiag + Toff + TCor + V [v3](R, θ1, θ2, φ) (10)

in which

Tstr = − 1

2μ

∂2

∂R2, (11)

Tdiag =(

1

2μR2+ Bv

CO

)[∂2

∂θ12 + cot θ1

∂

∂θ1

− 1

sin2 θ1(Jz − l2z)

2

]+

[1

2μR2+ BH2

]l

22

+ 1

2μR2[J

2 − 2(Jz − l2z)2 − 2Jz l2z], (12)

Toff = 1

2μR2[l2+ a −

1 + l2− a +1 ], (13)

TCor = − 1

2μR2[J+ a +

1 + J− a −1 + J+ l2+ + J− l2−], (14)

where

J± = Jx ± iJy, l2± = l2x ± il2y, (15)

a±1 = ± ∂

∂θ1− cot θ1(Jz − l2z). (16)

Here, the operators Jx , Jy , and Jz are the components of thetotal angular momentum operator J in the body-fixed frame,the z axis of the body-fixed frame lies along the Jacobi radialvector �R, and its x axis is in the plane that contains �R and theCO molecule. The above Hamiltonian contains full vibration-rotation coupling.

C. Calculating rovibrational energy levels

The rovibrational energy levels were calculated usingthe same approach as in Ref. 57. A sine discrete vari-able representation (DVR) grid79, 80 was used for the radialpart of the 4D Schrödinger equation. The angular part wasthen treated using parity-adapted rovibrational basis func-tions. In the parity-adapted angular finite basis representation(FBR), the kinetic energy terms have simple matrix elements.For the potential part, the matrix elements are not diagonalin the angular FBR basis. However, they could be calculatedin the grid representation by applying a three-dimensionaltransformation73 for the angles θ1, θ2, and φ, respectively,in which the potential energy matrix is diagonal. These in-tegrals need firstly the application of a transformation fromthe parity-adapted FBR to the DVR basis, then multiplicationby a diagonal potential matrix, and finally to be transformedback.13, 73 Gauss-Legendre quadrature was used for both theθ1 and θ2 angles, and Gauss-Chebyshev quadratures of thefirst kind were used to integrate φ for even and odd paritycases. The Lanczos algorithm was then used to calculate thero-vibrational energy levels by recursively diagonalizing theresulting discretized Hamiltonian matrix.81

III. ANALYTIC POTENTIAL ENERGY SURFACEFOR CO–H2

A. Potential energy function

The vibrational-averaged ab initio intermolecular poten-tial energies V [v](R, θ1, θ2, φ) for CO–H2 obtained fromEq. (7) were fitted to a generalization of the MLR potentialfunction form,55, 57, 82 which is written as

V MLR(R, θ1, θ2, φ)

= De(θ1, θ2, φ)

×[

1 − uLR(R, θ1, θ2, φ)

uLR(Re, θ1, θ2, φ)e−β(R,θ1,θ2,φ)·yeq

p (R,θ1,θ2,φ)

]2

,

(17)

in which De(θ1, θ2, φ) is the depth and Re ≡ Re(θ1, θ2, φ) theposition of the minimum on a radial cut through the potentialfor angles {θ1, θ2, φ}, while uLR(R, θ1, θ2, φ) is a functionwhich defines the (attractive) limiting long-range behaviourof the effective 1D potential along that cut as

V (R, θ1, θ2, φ) De(θ1, θ2, φ) − uLR(R, θ1, θ2, φ) + · · · .

(18)Since CO is polar, while H2 is non-polar, an appropriate func-tional form for uLR(R, θ1, θ2, φ) is

uLR(R, θ1, θ2, φ) = C4(θ1, θ2, φ)

R4+ C5(θ1, θ2, φ)

R5

+ C6(θ1, θ2, φ)

R6+ C7(θ1, θ2, φ)

R7

+ C8(θ1, θ2, φ)

R8, (19)

in which the long range coefficients Cn have also beenaveraged over the CO stretching coordinate C–O, and the

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164315-5 Li et al. J. Chem. Phys. 139, 164315 (2013)

denominator factor uLR(Re, θ1, θ2, φ) is that same functionevaluated at R = Re(θ1, θ2, φ).

The angle-dependent long-range coefficients for CO–H2 system with CO in its vibrational ground state(v = 0)are taken from the V98 potential,43 which scaled by the ra-tios of the experimental coefficient C000

6,exp83 to the calcu-

lated isotropic value.43 For CO in its vibrational excited state(v = 1), the angle-dependent long-range coefficients aretaken the same as those in ground state (v = 0).

The radial distance variable in the exponent in Eq. (17) isthe dimensionless quantity

yeqp (R, θ1, θ2, φ) = Rp − Re(θ1, θ2, φ)p

Rp + Re(θ1, θ2, φ)p, (20)

where p is a small positive integer which must be greaterthan the difference between the largest and smallest (inverse)powers appearing in Eq. (19), p > (8 − 4),55 and the expo-nent coefficient function β(R, θ1, θ2, φ) is a (fairly) slowlyvarying function of R, which is written as the constrainedpolynomial,

β(R, θ1, θ2, φ) = yrefp (R, θ1, θ2, φ)β∞(θ1, θ2, φ)

+ [1 − yref

p (R, θ1, θ2, φ)]

×N∑

i=0

βi(θ1, θ2, φ)yrefq (R, θ1, θ2, φ)i , (21)

whose behaviour is defined in terms of the two new radialvariables

yrefp (R, θ1, θ2, φ) = Rp − Rref

p

Rp + Rrefp and

yrefq (R, θ1, θ2, φ) = Rq − Rref

q

Rq + Rrefq (22)

in which, Rref ≡ fref × Re(θ1, θ2, φ). Although most previ-ous work with this model was performed using a single radialvariable to define the exponent coefficient function β(R, θ1,θ2, φ) (i.e., with q = p ) and with Rref = Re (i.e., with fref

= 1), it has recently been shown that use of fref > 1 and ofa separate smaller power q < p to define the radial variablein the power-series portion of Eq. (21) can lead more com-pact and robust potential functions.84 In the potential func-tion model used in the present work: p = 5, q = 3, andfref = 1.2.

The definition of yeqp (R, θ1, θ2, φ) and the algebraic struc-

ture of Eqs. (17) and (21) mean that

limR→∞

β(R, θ1, θ2, φ)

= limR→∞

{β(R, θ1, θ2, φ) · yeqp (R, θ1, θ2, φ)}

≡ β∞(θ1, θ2, φ)

= ln{2 De(θ1, θ2, φ)/uLR(Re, θ1, θ2, φ)}. (23)

The parameters De(θ1, θ2, φ), Re(θ1, θ2, φ), and the vari-ous exponent expansion coefficients β i(θ1, θ2, φ), all areexpanded in the form

F (θ1, θ2, φ) =∑l1,l2,l

Fl1l2lAl1l2l(θ1, θ2, φ), (24)

in which F = De , Re or β i , φ = φ1 − φ2, and l is thelabel associated with the vector sum of l1 and l2, with therange of values |l1 − l2| ≤ l ≤ |l1 + l2|. These three indicesmust also satisfy the restrictions that l2 is even, and l1 + l2+ l is even. The angular basis functions appearing here aredefined as

Al1l2l(θ1, θ2, φ)

=lmin∑

m=−lmin

(l1 l2 l

m −m 0

)Yl1,m(θ1, φ1)Yl2,−m(θ2, φ2), (25)

in which the quantity in large brackets is the Wigner 3jfactor,85 Yl, m(θ i, φi) are the normalized spherical harmonicfunctions, and lmin = min(l1, l2).

The presence of permanent dipole on CO, andquadrupole moments on CO and H2 means that the leadingterms in the expression for uLR(R, θ1, θ2, φ) are the electro-static dipole-quadrupole, and quadrupole-quadrupole interac-tion, whose (vibrationally averaged) coefficient may be writ-ten as

C4(θ1, θ2, φ) = − (√

105 ) μCO QH2A123(θ1, θ2, φ), (26)

C5(θ1, θ2, φ) = − (3√

70 ) QCO QH2A224(θ1, θ2, φ), (27)

in which μCO, QCO, and QH2are the vibrationally aver-

aged dipole and quadrupole moments of CO and H2, respec-tively. Our v-dependent values of μCO and QCO were ob-tained by averaging over the bond length dependence of thedipole and quadrupole moment function of CO reported byMaroulis.86

The vibrationally averaged dispersion coefficientsC6(7,8)(θ1, θ2, φ) may be expanded as

C6(7,8)(θ1, θ2, φ) =∑l1l2l

Cl1l2l

6(7,8) Al1l2l(θ1, θ2, φ).

An experimental value of the leading totally isotropic co-

efficient C0006,exp has been obtained from dipole oscillator

strength distributions by Kumar and Meath,83 but no angle-or stretching-dependent long-range coefficients have been re-ported for this system. Estimates of the dispersion coeffi-cients for (l1, l2, l) = (0, 0, 0) were therefore obtained fromthe theoretical calculations for the CO–H2.43 Our final val-ues of the coefficients for these angle-dependent terms werethen obtained by scaling these calculated coefficients by theratios of the “experimental” to the theoretical isotropic C6

coefficients:

Cl1l2l

6(7,8) = Cl1l2l

6(7,8),cal ×(C

0006,exp

/C

0006,cal

). (28)

Since our ab initio 5D PES incorporates the stretchingcoordinate of the CO monomer, the van der Waals interactionwill also include induction terms. Following Buckingham,87

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164315-6 Li et al. J. Chem. Phys. 139, 164315 (2013)

the coefficient of the R−6 induction term was taken to be

C6,ind(θ1, θ2, φ) =∑l1l2l

Cl1l2l

6,ind Al1l2l(θ1, θ2, φ)

= [μCO(r)

]2 ×[

αavH2

(A000 +√

5 A202)

+ (α‖H2

− α⊥H2

)

(√5

3A022 + 3√

5A220

−√

10√7

A222 + 2√

2√35

A224

)], (29)

in which the leading factor on the right-hand side is the vibra-tional average of the square of the CO dipole moment, whileα

‖H2

, α⊥H2

, and αavH2

are, respectively, the parallel, perpendicular,and isotropic-average polarizabilities of H2. The latter weredefined by the vibrationally averaged values for ground-stateH2 reported by Bishop and Cheung,88 while the former wascalculated using the stretching-dependent CO dipole momentreported by Maroulis.86

Finally, while the v-dependence of the angle-dependentdispersion terms was neglected, that for the leading isotropiccoefficient was assumed to scale as the isotropic averagepolarizability of CO αav

CO(r) = [α‖CO(r) + 2 α⊥

CO(r)]/3 . Us-ing the r–dependent polarizabilities for CO reported byMaroulis,86 this yields

C0006,disp(v = 1) = C

0006,disp(v = 0) × 〈ψv=1

∣∣αavCO(r)

∣∣ψv=1〉〈ψv=0

∣∣αavCO(r)

∣∣ψv3=0〉.

(30)

B. Least-squares fits

To commence any nonlinear least-squares fit, it is neces-sary to have realistic initial trial values of the fitting parame-ters. In the present case of fits to the 4D Morse/Long-Range(4D-MLR) form of Eq. (17), they were obtained in the follow-ing manner. First, a fit to the ordinary 1D MLR form (depend-ing only on R) was performed for all distinct combinations ofθ1, θ2, and φ, using program betaFIT.89 This involved someexperimentation to ascertain the most appropriate choice forthe integer parameter p and q and the factor fref appear-ing in the definitions of the radial variables y

eqp (R; θ1, θ2, φ),

yrefp (R; θ1, θ2, φ), and yref

q (R; θ1, θ2, φ) of Eqs. (20) and (22),and for the order N of the exponent polynomial of Eq. (21). Aspointed out above, the present potential function model usedp = 5, q = 3, and fref = 1.2, and the exponent polynomial or-der was N = 4 . The resulting values of De(θ1, θ2, φ), Re(θ1,θ2, φ), and of β i(θ1, θ2, φ) (for i = 0 − N) were then fit-ted to Eq. (24), and the resulting expansion coefficients Fl1l2l

used as starting parameters in the global 4D fits of the vibra-tionally averaged potential energies to Eq. (17). In the fol-lowing parts of the present paper, the vibrationally averaged4D “Morse/long-range” potential energy surfaces calculatedat CCSD(T) level with two-point extrapolation are denoted asVMLR, and the VMLR potential energy surfaces which further-more incorporated the correlation energies from the triple and

quadruple excitations calculated at CCSDT(Q)/aug-cc-pVDZare denoted as VMLRQ.

In the final 4D fits, the input ab initio energies wereweighted by assigning uncertainties of ui = 0.1cm−1 topoints in the attractive well region where V (R, θ1, θ2, φ)≤ 0.0 cm−1, and ui = [V (R, θ1, θ2, φ) + 5.0]/50.0cm−1 tothose in the repulsive wall region where V (R, θ1, θ2, φ)> 0.0 cm−1. Using these weights, our final 196-parametersfits to the 30 206 vibrationally averaged interaction energiesfor V (R, θ1, θ2, φ) < 500 cm−1 yielded dimensionless root-mean-square (RMS) residual discrepancies of only 0.087 and0.082 cm−1 for v = 0 and v = 1 of VMLR potentials, and0.095 and 0.087 cm−1 of VMLRQ potentials, respectively.More than one-third of (79/196) of those fitting parame-ters are required to define De(θ1, θ2, φ), A see Table I, 56to define Re(θ1, θ2, φ), and 37, 16, 7, 5, and 3 to defineβ i(θ1, θ2, φ) for i = 0–4, respectively. At the resolution of

TABLE I. Expansion coefficients Dl1,l2,l

e [cm−1], defining our 4-dimensional vibrationally averaged potential energy surfaces for 12C16O(v = 0)–H2.

D0,0,0e 45.266 D

0,2,2e 2.19 D

0,4,4e 1.07

D1,0,1e − 5.098 D

1,2,1e − 1.49 D

1,4,3e 0.06

D2,0,2e − 17.74 D

1,2,3e − 9.81 D

1,4,5e 1.03

D3,0,3e − 2.9 D

2,2,0e 1.29 D

2,4,2e 0.13

D4,0,4e 4.13 D

2,2,2e 10.32 D

2,4,4e 1.94

D5,0,5e 5.62 D

2,2,4e 180.02 D

2,4,6e 4.98

D6,0,6e − 0.05 D

3,2,1e 0.84 D

3,4,3e 0.06

D7,0,7e − 2.64 D

3,2,3e 6.18 D

3,4,5e 1.05

D8,0,8e − 0.5 D

3,2,5e 73.0 D

3,4,7e 3.04

D9,0,9e 0.78 D

4,2,2e 0.07 D

4,4,2e 0.16

D10,0,10e 0.37 D

4,2,4e 7.04 D

4,4,4e 0.57

D11,0,11e − 0.15 D

4,2,6e − 5.09 D

4,4,6e 2.16

D12,0,12e − 0.12 D

5,2,3e − 0.19 D

4,4,8e 16.31

D5,2,5e 0.76 D

5,4,3e 0.16

D5,2,7e − 18.64 D

5,4,5e 0.38

D6,2,4e 0.28 D

5,4,7e 1.64

D6,2,6e − 1.79 D

5,4,9e 14.08

D6,2,8e − 3.23 D

6,4,2e 0.06

D7,2,5e 0.21 D

6,4,6e 0.31

D7,2,7e − 0.67 D

6,4,8e 1.05

D7,2,9e 5.35 D

6,4,10e 4.39

D8,2,6e − 0.05 D

7,4,3e − 0.03

D8,2,8e 0.19 D

7,4,7e − 0.07

D8,2,10e 2.81 D

7,4,9e 0.32

D9,2,7e − 0.14 D

7,4,11e − 2.23

D9,2,9e 0.33 D

8,4,10e − 0.3

D9,2,11e − 1.0 D

8,4,12e − 2.26

D10,2,8e − 0.1 D

9,4,11e − 0.2

D10,2,10e 0.08 D

10,4,6e 0.06

D10,2,12e − 1.02 D

10,4,8e − 0.1

D11,2,9e 0.09 D

10,4,12e − 0.2

D11,2,11e − 0.09 D

12,4,8e − 0.12

D12,2,10e 0.24

D12,2,12e − 0.08

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164315-7 Li et al. J. Chem. Phys. 139, 164315 (2013)

ΔV(θ

1, θ 2,

φ) /

cm-1

-100

-50

0

50

(0°, 0°)

(0°, 90°)

(180°, 0°)

(90°, 0°)

(90°, 90°) (180°, 90°)

(θ1, θ2, φ=0°)ΔV

(θ1,

θ 2, φ)

/ cm

-1

2 3 4 5 6 7 8-100

-50

0

50

(0°, 0°)

(0°, 90°)

(180°, 0°)

(90°, 0°)

(90°, 90°) (180°, 90°)

(θ1, θ2, φ=90°)

R / Å

FIG. 2. Vibrational averaged ab initio interaction energies (points) along cutsthrough the analytic 4D VMLRQ potential energy surface for CO–H2 at variousrelative orientations.

Fig. 2, the resulting fitted potential passes through all of the(vibrationally averaged) ab initio points shown. The resultingsets of potential parameters and a FORTRAN subroutine forgenerating these potentials may be obtained from the authorsor from the journal’s supplementary data archive.90 Our useof the sequential rounding and refitting procedure of Ref. 91means that the parameter sets are relatively compact.

IV. RESULTS AND DISCUSSION

A. Features of the four-dimensional potentialenergy surface

Figure 3 shows how the well depth of our fitted, vibra-tionally averaged, 4D VMLRQ ground-state potential energysurface for CO(v=0)–H2 system depends on θ1 and θ2 whenφ is optimized to minimize the energy for each (θ1, θ2). Asseen there, two equivalent global minima with well depth of93.33 cm−1 occur for the linear geometries (θ1 = 0, θ2 = 0◦)or (θ1 = 0, θ2 = 180◦) by symmetry of H2, while both withR = 4.188 Å and φ = 0◦, where C atom pointing toward theH2 molecule. Along the lowest-energy isomerization path be-tween the two equivalent global minima, there exists a tran-sition state with a barrier of height 54.89 cm−1 located at(θ1 = 55.8, θ2 = 43.8◦, φ = 90.0◦) with R = 3.960 Å.Figure 3 also shows two equivalent local minima with ener-gies of −74.36 cm−1 appear at collinear geometries where O

060

120

-100

-80

-60

-40

-20

0

060

120180

-100

-80

-60

-40

-20

0

− De(θ1, θ2 ; φ optimized)

−De / cm-1

θ2(°)

θ1(°)

FIG. 3. Minimum energy on our vibrationally averaged 4D VMLRQ PES forCO(v = 0)–H2 as a function of angles θ1 and θ2, for optimized values ofφ and R.

atom pointing toward the H2 molecule, with (θ1 = 180, θ2

= 0◦) or (θ1 = 180, θ2 = 180◦), and R = 3.789Å φ = 0◦.Along the lowest-energy isomerization path between the twoequivalent local minima, there is a saddle point with a barrierof height 28.48 cm−1 located at (θ1 = 113.2, θ2 = 124.6◦,φ = 90.0◦) with R = 3.582 Å.

Figure 3 also illustrates two types of minimum energypaths joining these local minima to the global minima. Thefirst is a low barrier path with a barrier of height 24.89 cm−1

relative to the global minima located at (θ1 = 106.9,θ2 = 56.8◦) with R = 3.473 Å and φ = 0◦, which is moreclearly shown in the middle panel of Figure 4. Along this paththe molecules remain co-planar ( φ = 0◦ ) and θ1 increaseswhile θ2 decreases (or vice versa), so that the H2 molecule ro-tates through 180◦ in a direction counter to the rotation of theH2 relative to the CO axis as it moves to the collinear arrange-ment. Figure 5 clearly shows the relative rotational directionsfor vector O–C (blue) and vector of H(2)–H(1) (red) along theminimum energy path in cylindrical coordinates on our vi-brationally averaged 4D VMLRQ PES for CO(v = 0)–H2 withφ = 0◦. The second are two high barrier paths, with barrierheights of 54.89 cm−1 and 47.45 cm−1 relative to the globalminima, respectively. The transition states on these two pathsare located at the same positions as those two saddle pointsalong the isomerization paths between two global and two lo-cal minima, respectively.

Figure 6 shows how the radial positions of the minimumenergy depend on θ1 and θ2 when φ is optimized at everypoint. The dotted curves seen there indicate configurations atwhich the optimum value of switches abruptly between 0◦ and90◦. As may be expected, contours which cross these dottedcurves show small discontinuities at these switchover points.Nonetheless, the fact that the structure seen here is somewhat

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164315-8 Li et al. J. Chem. Phys. 139, 164315 (2013)

Δ De (θ

1) / c

m-1

-2

0

2VMLRQ

VMLR

V12

Δ De(θ1)−D

e (θ

1) / c

m-1

-90

-80

-70

VMLRQ

VMLR

V12

V04

−De(θ1) saddle point (68.44 cm-1)

5.92 cm-1

24.89 cm-1

global minimum (93.33 cm-1)

local minimum (74.36 cm-1)

θ1(°)

1

Re (θ

1) / c

m-1

0 30 60 90 120 150 180

3.4

3.6

3.8

4.0

4.2

4.4VMLRQ

VMLR

V12

V04

Re(θ1)

FIG. 4. Energy (upper) and radial position (lower) along the minimum en-ergy path on our vibrationally averaged 4D VMLRQ PES for CO(v = 0)–H2as functions of angle θ1 for optimized values of θ2, φ, and R.

simpler than that seen in Figure 3, indicates why the descrip-tion of Re(θ1, θ2, φ), requires fewer parameters 56 versus 70are require to define De(θ1, θ2, φ). One of the nice featuresof the generalized MLR form is the fact that these two phys-

x = Re cos (θ1) / Å

y =

Re si

Ån

(θ1)

/

-4 -2 0 2 4-1

0

1

2

3

4

O C H(1)H(2)

FIG. 5. Rotational directions for vector O–C (blue) and vector of H(2)–H(1)

(red) along the minimum energy path in cylindrical coordinates on our vibra-tionally averaged 4D VMLRQ PES for CO(v = 0)–H2 with φ = 0.

3.9

3.7

4.2

3.7

4.14.0

4.1

3.7

3.6

3.5

3.6

3.5

3.8

3.9

3.6

3.7

3.8

4.5

4.0

3.8

3.7

3.6

4.6

4.5

4.3

4.4

3.9 3.8 3.7

4.4

4.3

4.2

3.9

θ1(°)

θ 2(°)

0 30 60 90 120 150 1800

30

60

90

120

150

180 Re(θ1,θ2; φ optimized)

φopt = 90°

φopt = 0°

φopt = 90°

FIG. 6. Radial position of the minimum on our vibrationally averaged 4DVMLRQ PES for CO(v = 0)–H2 as functions of angle θ1 and θ2 for optimizedvalues of φ and R.

ically meaningful quantities,De(θ1, θ2, φ) and Re(θ1, θ2, φ),which are directly determined by the fit, incorporate most ofthe basic structural information about our 4D VMLRQ surfaces.

The geometries and energies of these global minima andsaddle points are summarized, and compared with previousliterature results in Table II. As shown in Table II, all the sta-tionary points (the global minimum, saddle point, and localminimum) of VMLRQ(v = 0) are lower than those correspond-ing points on VMLR(v = 0) surface due to the high-order ofcorrelation energies correction of �V

T(Q)int for VMLRQ(v = 0)

potential. These differences are anisotropical, with 2.68, 0.92,and 0.69 cm−1 for the global minimum, saddle point, andlocal minimum, respectively, and this is possible an impor-tant factor affecting the final spectra transitions. Comparedbetween VMLRQ(v = 0) and recent new V12(v = 0), the dif-ferences at the global minimum, saddle point, and localminimum are reduced to −0.77, 0.04, and 0.62 cm−1. Thosedifferences are originated from different methods of calcula-tions, from different basis sets, and from a different treatmentof nonrigidity effect of H2. The comparison among VMLRQ

(v = 0), VMLR(v = 0), V12(v = 0), and V04(v = 0) along theminimum energy paths are illustrated in the middle panel ofFigure 4, and the effect for the spectra transitions will bechecked later on. From Table II, one can see that all the posi-tions of the stationary points on our VMLRQ are in good agree-ment with those values from recent ab initio surfaces for thissystem calculated by Jankowski et. al.,44, 52 with radial differ-ence smaller than 0.01 Å and angle smaller than 1◦, whichalso clearly indicated in the lower panel of Figure 4, wherethe radial position along the minimum energy paths on recentab initio surfaces are indistinguishable.

For the vibrationally averaged excited-state (v = 1) sur-face, the contours plots look almost the same as those for

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164315-9 Li et al. J. Chem. Phys. 139, 164315 (2013)

TABLE II. Properties of stationary points of the CO-H2 potential energy surface, and comparisons with results for previously reported surfaces. All entriesare given as {R [Å] θ◦

1 , θ◦2 , �V [cm−1]} with φ = 0◦.

Gobal minimum Saddle point Local minimum Ref.

VMLRQ(v = 0) {4.188, 0.0, 0.0, −93.33} {3.473, 106.9, 56.8, −68.44} {3.789, 180.0, 0.0, −74.36} PresentVMLRQ(v = 1) {4.196, 0.0, 0.0, −91.98} {3.469, 105.3, 57.3, −69.01} {3.790, 180.0, 0.0, −75.37} PresentVMLR(v = 0) {4.198, 0.0, 0.0, −90.65} {3.477, 106.5, 57.2, −67.52} {3.792, 180.0, 0.0, −73.67} PresentVMLR(v = 1) {4.206, 0.0, 0.0, −89.34} {3.472, 104.9, 57.8, −68.08} {3.793, 180.0, 0.0, −74.69} PresentV12(v = 0) {4.186, 0.0, 0.0, −94.10} {3.472, 108.1, 55.9, −68.40} {3.793, 180.0, 0.0, −73.74} 52V12(v = 1) {4.194, 0.0, 0.0, −92.77} {3.465, 105.9, 57.1, −68.96} {3.794, 180.0, 0.0, −74.77} 52V04 {4.191, 0.0, 0.0, −93.05} {3.794, 180.0, 0.0, −72.74} 44V98 {4.106, 0.0, 0.0,−109.27} {3.762, 180.0, 0.0, −84.57} 43

{4.212, 0.0, 0.0, −84.45} {3.852, 180.0, 0.0, −66.78} 95{4.228, 0.0, 0.0, −85.98} {3.837, 180.0, 0.0, −59.85} 96{4.233, 0.0, 0.0, −74.50} 97

the ground state (v = 0), and as shown in Table II, the po-sitions and energies of the stationary points are only slightlyshifted. Because the energy difference between vibrationallyaveraged ground-state (v = 0) and excited-state (v = 1) sur-face is related to the band origin shifts, it is interesting to seethe differences along the minimum energy path and comparedamong the different model potentials. As shown in the up-per panel of Figure 4, the different potential curves along theminimum energy path, among the three �VMLRQ, �VMLR, and�V12 model potentials, are indistinguishable, with energy dif-ference smaller than 0.02 cm−1, which indicated that the bandorigin shifts will have very small effect from different modelpotentials and will be globally tested by our bound state cal-culations and Path Integral Monte Carlo simulations for largerdoped clusters.

B. Bound states and band origin shiftsfor CO–H2 dimer

The rovibrational energy levels of CO–H2 were calcu-lated using the radial DVR and parity-adapted angular FBRmethods described in Secs. II B and II C. Because of thesymmetry properties associated with P and l2, there exist foursymmetry blocks, and the rovibrational energy levels for eachblock could be calculated separately. An 80-point sine-DVRgrid range from 3.0 to 20.0 bohrs was used for the radial Rstretching coordinate, and 27 and 19 associated Legendre ba-sis functions were used for the angular coordinates θ1 and θ2,respectively. The integration over θ1 and θ2 used 32 and 24Gauss-Legendre quadrature points, respectively, and that overφ used 52 equally spaced points in the range [0, 2π ].

The calculated intermolecular rovibrational energy lev-els compared with corresponding experimental values for theground (v = 0) and first excited (v = 1) states of para H2–CO are listed in Table III and ortho H2–CO in Table S1 ofthe supplementary material,90 respectively. The rovibrationalenergy levels may be labeled by the six quantum numbers: v,J, l1, l2, l12, and l, where v is the stretch quantum number ofCO, J is the total angular momentum, l1, l2, l12, and l are theeigenvalues of the l1, l2, l12, and l operators, respectively. Itis known that the rotation of H2 in the complex is dominatedby l2 = 0 terms for para H2 and by l1 = 1 for ortho H2, due

to the large spacings between the rotational energy levels ofmolecular hydrogen. For para H2–CO, when l2 = 0, from therules of the angular momentum coupling, the l12 is equal tol1. Therefore, the set of indices used for a unique labeling ofthe components of the rovibrational for para H2–CO complexreduced to (v, J, P, l1, l). However, For ortho H2–CO, angu-lar momentum l2 with the eigenvalue l2 = 1 can be coupledwith l1 in up to three different ways characterized by l12.Thus,unlike inthe para-H2–CO case for the ortho H2–CO, we alsoneed l12 for a unique labeling of the rovibrational energy lev-els. Same labeling method have been used in previous stud-ies for H2–CO complex.43, 44, 52, 53 Following the approach ofRef. 92, our calculations for para H2-CO complexes used aneffective H2-molecule inertial rotational constant BH2 calcu-lated from the experimental l2 = 0 → 2 level spacing, whileour ortho H2–CO calculations used a value of BH2 defined bythe l2 = 1 → 3 monomer level spacing.

Table III lists the observed rovibrational levels andenergy differences between the theoretically predictionsand corresponding experimental observations for para H2–CO(v=0) and para H2–CO(v=1), respectively. All theoret-ical rovibrational energies are given relative to the groundstate energies for v = 0 or v = 1, respectively. On our VMLRQ

surfaces, ground state energies are equal to −19.371 and−19.550 cm−1, which are in good agreement with the val-ues of −19.440 and −19.616 cm−1 calculated on recent V12

surfaces53 with the discrepancies smaller than 0.07 cm−1.While on our VMLR surfaces, the calculated ground state en-ergies of −18.654 and −18.833 cm−1 for v = 0 or v = 1, re-spectively, which are 0.717 and 0.783 cm−1 higher than thosecorresponding values from our VMLRQ surfaces. Because ourcode does not allow to calculate resonance states, total of 39bound state energies from J = 0 to J = 6 are predicted. Asseen in columns 4 and 7 of Table III, the differences betweenthe theoretical predicted values yielded by our 4D-VMLRQ sur-faces and experimental values for v = 0 or v = 1 states arevery small, and the RMS differences are only 0.006 and 0.008cm−1 for v = 0 or v = 1 states, which are almost in the sameaccuracy as the those differences of 0.005 and 0.007 cm−1

yielded from V12 surfaces. While on VMLR surfaces, as shownin columns 5 and 10 of Table III for v = 0 or v = 1, thedifferences with experiment are then significantly increased,

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164315-10 Li et al. J. Chem. Phys. 139, 164315 (2013)

TABLE III. The calculated rovibrational energy levels(cm−1) for paraH2–CO complex from our vibrationally averaged 4D-VMLR and VMLRQ PES, comparisonwith experiment and previous theory predictions. D0 is the dissociation energy and �v0 is the band origin shift.

paraH2−CO(v = 0) paraH2−CO(v = 1)

Levels Diff.(Calc.-Obs.) Diff.(Calc.-Obs.)

J P l1 l Obs. VMLRQ VMLRQ VMLR V12 Obs. VMLRQ VMLRQ VMLR V12

0 e 0 0 0.000 0.000 0.000 0.000 0.000 0.000 0.0000 0.000 0.000 0.0000 e 1 1 7.079 7.083 0.004 − 0.031 0.004 7.056 7.0568 0.001 − 0.036 − 0.0040 e 2 2 15.169 15.168 − 0.001 − 0.091 0.009 15.121 15.1185 − 0.003 − 0.086 0.0011 f 1 1 4.090 4.092 0.002 0.032 − 0.005 4.067 4.0647 − 0.002 0.030 − 0.0081 f 2 2 15.665 15.666 0.001 − 0.030 − 0.011 15.570 15.5677 − 0.002 − 0.033 − 0.0151 e 0 1 1.054 1.053 − 0.001 − 0.009 0.001 1.059 1.0536 − 0.005 − 0.014 − 0.0041 e 1 0 3.618 3.621 0.003 0.035 − 0.004 3.596 3.5941 − 0.002 0.032 − 0.0091 e 1 2 8.485 8.487 0.002 − 0.041 0.004 8.463 8.4619 − 0.001 − 0.045 − 0.0031 e 2 1 13.660 13.663 0.003 − 0.026 − 0.002 13.583 13.5823 − 0.001 − 0.030 − 0.0101 e 2 3 17.913 17.923 0.010 − 0.139 0.001 17.872 17.8744 0.002 − 0.136 − 0.0062 f 1 2 6.266 6.265 − 0.001 0.015 − 0.003 6.246 6.2405 − 0.006 0.012 − 0.0082 f 2 1 11.690 11.693 0.003 0.042 − 0.004 11.600 11.5982 − 0.002 0.040 − 0.0082 f 2 3 18.342 18.339 − 0.003 − 0.052 − 0.010 18.252 18.2460 − 0.006 − 0.055 − 0.0132 e 0 2 3.148 3.145 − 0.003 − 0.028 0.002 3.153 3.1457 − 0.007 − 0.033 − 0.0022 e 1 1 5.014 5.017 0.003 0.026 − 0.003 4.994 4.9922 − 0.002 0.023 − 0.0092 e 1 3 11.097 11.096 − 0.001 − 0.057 0.009 11.059 11.0550 − 0.004 − 0.054 0.0042 e 2 0 11.367 11.369 0.002 0.031 − 0.010 11.296 11.2947 − 0.001 0.023 − 0.0152 e 2 2 14.807 14.808 0.001 − 0.027 0.000 14.731 14.7295 − 0.002 − 0.028 − 0.0053 f 1 3 9.490 9.487 − 0.003 − 0.010 − 0.001 9.473 9.4655 − 0.008 − 0.013 − 0.0053 f 2 2 14.079 14.080 0.001 0.020 − 0.003 13.990 13.9861 − 0.004 0.017 − 0.0083 f 2 4 22.111 22.104 − 0.007 − 0.091 − 0.007 22.029 22.0186 − 0.010 − 0.093 − 0.0113 f 3 1 23.164 23.168 0.004 0.040 − 0.003 22.972 22.9714 − 0.001 0.038 − 0.0073 e 0 3 6.248 6.242 − 0.006 − 0.060 0.005 6.254 6.2424 − 0.012 − 0.066 0.0013 e 1 2 7.293 7.295 0.002 0.015 − 0.004 7.277 7.2752 − 0.002 0.013 − 0.0093 e 2 1 12.933 12.937 0.004 0.012 0.000 12.845 12.8438 − 0.001 0.008 − 0.0063 e 1 4 14.821 14.815 − 0.006 − 0.095 0.001 14.811 14.8022 − 0.009 − 0.099 − 0.0043 e 2 3 17.501 17.500 − 0.001 − 0.040 0.005 17.427 17.4223 − 0.005 − 0.042 − 0.0014 f 1 4 13.709 13.702 − 0.007 − 0.046 0.002 13.698 13.6860 − 0.012 − 0.049 − 0.0034 f 2 3 17.403 17.401 − 0.002 − 0.012 − 0.001 17.317 17.3098 − 0.007 − 0.015 − 0.0064 e 0 4 10.272 10.258 − 0.014 − 0.123 0.013 10.260 10.2414 − 0.019 − 0.128 0.0094 e 1 3 10.517 10.520 0.003 0.017 − 0.008 10.527 10.5255 − 0.002 0.015 − 0.0144 e 2 2 15.255 15.257 0.002 − 0.017 0.000 15.172 15.1687 − 0.003 − 0.021 − 0.0064 e 1 5 19.411 19.401 − 0.010 − 0.146 0.003 19.411 19.3970 − 0.014 − 0.150 − 0.0045 f 1 5 18.831 18.819 − 0.012 − 0.095 0.006 18.829 18.8121 − 0.017 − 0.098 0.0005 f 2 4 21.650 21.642 − 0.008 − 0.058 0.001 21.567 21.5552 − 0.012 − 0.060 − 0.0045 e 1 4 14.524 14.517 − 0.007 − 0.084 0.003 14.510 14.4979 − 0.012 − 0.091 − 0.0025 e 0 5 15.273 15.261 − 0.012 − 0.104 0.006 15.297 15.2810 − 0.016 − 0.103 0.0005 e 2 3 18.426 18.423 − 0.003 − 0.058 0.002 18.347 18.3411 − 0.006 − 0.061 − 0.0046 e 1 5 19.401 19.391 − 0.010 − 0.114 0.001 19.402 19.3878 − 0.014 − 0.120 − 0.004

RMSD 0.006 0.064 0.005 0.008 0.065 0.007D0 19.371 18.654 19.440 19.550 18.833 19.616�v0 − 0.179 − 0.179 − 0.179 − 0.176

yielding the RMS discrepancies of 0.064 and 0.065 cm−1, re-spectively. So big discrepancies are almost 10 times worsethan the values yielding from our VMLRQ surfaces, only be-cause our VMLR surfaces missing the term of �V

T(Q)int for elec-

tron correlation energies from the triple and quadruple excita-tions. However, comparable RMS differences between VMLRQ

and V12, indicate that extra contributions from V12 surface,such as nonrigidity effect of H2, core-valence electron corre-lation, and larger basis sets extrapolation have negligible ef-fects on the calculated rovibrational levels for para-H2–COcomplex.

For the ortho H2–CO complex, whose level energies areexpressed relative to the jH 2 =1 dissociation channel, wefind a total of 103 bound intermolecular vibrational states onour 4D VMLRQ surfaces with energies lower than its asymptotelimit (l2 = 1 at 118.644 cm −1), three times more than werefound for the para H2–CO2 complex. All the relative ener-gies are positive and listed in Table S1 of the supplementarymaterial,90 with the ground states at 96.802 and 96.607 cm−1

for v = 0 and 1, respectively. The only good quantum num-bers are the overall angular momentum J and parity P = eor f. The CO rotational quantum number l1, the number l12

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164315-11 Li et al. J. Chem. Phys. 139, 164315 (2013)

describing the coupling of l1 and l2, and the end-over-endnumber l are all approximate. nJ, P simply numbers consec-utive states for each J, P.

As shown in the last row of Table III, the calculatedband origin shift predicted by our 4D-VMLRQ surfaces is �v0

= −0.179 cm−1 for para H2–CO, which is exact the samevalue of the experimental observation, and slightly differentfrom the predicted value yielding on the V12 surfaces.53 Itis also interesting to notice that the same the band originshift of −0.179 cm−1 is yielded from our 4D-VMLR surfaces,which indicate that missing the term of �V

T(Q)int for electron

correlation energies from the triple and quadruple excitationshave negligible effects on the calculated band origin shift forpara H2–CO complex. This result is also consistent with theresults discussed in Sec. IV A and shown in the upper panel ofFigure 4, the different potential curves along the minimum en-ergy path, have tiny difference between �VMLRQ and �VMLR.For para H2–CO, the calculated band origin shift on our 4D-VMLRQ surfaces is �v0 = −0.195 cm−1, which is 0.017 cm−1

different from the predicted value of −0.178 cm−1 calculatedfrom V12 surfaces.53

C. Predicted infrared spectra for CO–H2 dimer

For para H2–CO,infrared v = 0 → 1 transition frequen-cies calculated from our vibrationally averaged 4D-VMLRQ arelisted Table IV, and compared with experiment and with pre-vious theoretical predictions. The rotational levels were as-signed using the labels J, l1, and l, where J is the total an-gular momentum, l1 denotes the angular momentum of theCO, and l denotes the end-over-end rotation of the complex.Table IV expresses all the infrared transition energies relativeto the band origin 2143.092 cm−1 of para H2–CO complex.To list all the transitions more compact, the “mirror-image”pairs of transitions involving the same set of rotational quan-tum numbers for the complex, the “Obs.,upper” (J ′

l′1l′← J ′′

l′′1 l′′)and the “Obs.,lower” (J ′′

l′′1 l′′ ← J ′l′1l′

), are listed on the sameline of Table IV, as used in Ref. 50. Columns 3 and 8 show thetransition energies yielded by our vibrationally averaged 4D-VMLRQ potential energy surfaces, which are seen to agree verywell with the experimental values shown in columns 2 and7.50 The differences seen in columns 4 and 9 are very small,and the RMS discrepancy for total 126 bound state transitionsis only 0.007 cm−1, which are slightly smaller than the valueof 0.010 cm−1 obtained from V12 surfaces. However, withoutthe correction term of �V

T(Q)int for electron correlation ener-

gies, the transition energies are yielded from VMLR surfaces,the differences with experiment50 are then significantly in-creased as is shown in columns 5 and 10 of Table IV, yieldingRMS discrepancy of 0.056 cm−1, eight times worse than thatobtained from VMLRQ surfaces.

For ortho H2–CO, the calculated infrared transition fre-quencies expressed relative to the band origin 2143.272 cm−1

of free CO are given in Table S290 of the supplementary mate-rial. The lower and upper states involved in the transitions arelabeled with the set of parameters (J, P, nJ, P), where J andP are the total angular momentum and parity, respectively,nJ, P numbers consecutive states for each J, P. Columns 3 and

8 of Table S2 given in the supplementary material90 list thevalues calculated from our vibrationally averaged 4DVMLRQ

PESs, which agree very well with the experimental valuesshown in Columns 2 and 7 from Ref. 53. The differencesseen in columns 4 and 8 are very small, the RMS discrep-ancy of 0.023 cm−1 for total of 179 bound state transitions,is only 0.005 cm−1 larger than that value of 0.018 cm−1 ob-tained from V12 potential energy surfaces.

D. Vibrational band shifts of CO in (para-H2)N clusters

The vibrational frequency shifts of chromophoremolecule in He or (para-H2)N clusters were simulated byusing the path-integral Monte Carlo (PIMC) method,35, 57 inwhich the finite-temperature PIMC code with the rotation ofdopant and the bosonic exchange developed by Blinov andRoy93, 94 was used to calculate the density matrix and obtaincanonical averages, and first-order perturbation theory wasused to estimate the vibrational frequency shift.35 The presentimplementation closely follows our the previous work.34–36, 57

When the CO molecule is embedded in (para-H2)N clus-ters, its vibrational frequencies will be shifted from the valuesfor the free CO molecule. Based on first-order perturbationtheory, the fundamental band origin frequency shifts of CO insuch clusters can be estimated by

�νN0 ≡ ⟨

�V totparaH2−CO

⟩β

=∫ ∫

ρ(R, θ1)�VparaH2−CO(R, θ1)dθ1dR (31)

in which, �VparaH2−CO(R, θ1) is the difference potentialbetween effective 2D PESs for para H2–CO(v = 0) andpara H2–CO(v = 1), which are obtained by performing areduced-dimension adiabatic-hindered-rotor average over theorientations of the H2 moiety proposed by Li, Roy, andLe Roy.58 ρ(R, θ1) is the normalized density distributionof the solvent para H2 in the clusters, obtained from finite-temperature PIMC simulations. 512 translational and 128 ro-tational time slices were used in the present simulations,which were large enough to converge the results, and testedin our previous study of para H2 clusters.29, 34 Parameters ofm = 16 defines the interval for selecting a random numberof m (1 ≤ m ≤ m) consecutive beads in a world line, andthe adjustable parameter C range from 0.085 to 0.75 for dif-ferent cluster size controls the relative statistics of diagonaland off-diagonal configuration spaces in the worm algorithmwere chosen to keep the acceptance ratio between 0.25 and0.55. Figure 7 presents the simulated values of �νN

0 on thedifferent model potentials, compared with the experimentaldata.34 The listing of the calculated frequency shifts shown inFigure 7 may be obtained from the supplementary material.90

The statistical errors are on the order of 10−4 cm−1 and thusthe error bars for the data are smaller than the symbols andare not shown here.

Figure 7 shows a monotonic redshifts (open circle) for(para-H2)N–CO clusters calculated on the VMLRQ PESs de-pend on the number of para-H2 in the complex. For the small-est clusters, the vibrational band shift decreases almost lin-early from N = 1 to 3, but becomes increasingly nonlinear

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164315-12 Li et al. J. Chem. Phys. 139, 164315 (2013)

TABLE IV. Predicted infrared transition frequencies for para H2−CO around observed value of 2143.092 cm−1 from our vibrationally averaged 4D-VMLR

and VMLRQ PES, comparison with experiment and previous theory predictions.

Obs., upper(J ′l′1l′ ← J ′′

l′′1 l′′ ) Obs., lower(J ′′l′′1 l′′ ← J ′

l′1l′ )

TransitionsDiff.(Calc.-Obs.) Diff.(Calc.-Obs.)

J ′l′1l′ − J ′′

l′′1 l′′ Obs. VMLRQ VMLRQ VMLR V12 Obs. VMLRQ VMLRQ VMLR V12

422 − 314 0.350 0.353 0.003 0.075 − 0.006 − 0.444 − 0.454 − 0.010 − 0.081 − 0.004515 − 523 0.403 0.389 − 0.014 − 0.039 − 0.002 − 0.484 − 0.478 0.006 0.033 − 0.010110 − 202 0.448 0.449 0.001 0.060 − 0.011 − 0.465 − 0.475 − 0.011 − 0.068 0.002101 − 000 1.054 − 1.054 − 1.053 0.001 0.009 − 0.001415 − 523 0.985 0.974 − 0.011 − 0.091 − 0.006 − 1.064 − 1.060 0.004 0.085 − 0.007303 − 211 1.240 1.226 − 0.014 − 0.092 0.004 − 1.255 − 1.249 0.005 0.083 − 0.013212 − 211 1.224 − 1.272 − 1.273 − 0.001 0.008 − 0.006422 − 414 1.462 1.467 0.005 0.025 − 0.007 − 1.558 − 1.571 − 0.013 − 0.031 − 0.002321 − 213 1.748 1.748 0.000 0.065 − 0.015 − 1.874 − 1.882 − 0.007 − 0.066 0.004202 − 101 2.100 2.092 − 0.007 − 0.024 − 0.004 − 2.088 − 2.091 − 0.003 0.014 − 0.007313 − 312 2.181 2.171 − 0.010 − 0.028 − 0.002 − 2.213 − 2.212 0.001 0.023 − 0.008404 − 312 2.967 2.947 − 0.021 − 0.143 0.013 − 2.995 − 2.983 0.012 0.136 − 0.022111 − 101 3.014 3.011 − 0.002 0.038 − 0.010 − 3.031 − 3.038 − 0.007 − 0.045 0.001303 − 202 3.098 − 3.095 − 3.096 − 0.001 0.027 − 0.007212 − 202 3.098 3.096 − 0.002 0.040 − 0.010 − 3.112 − 3.120 − 0.007 − 0.048 0.000414 − 413 3.181 3.166 − 0.015 − 0.066 0.005 − 3.181 − 3.176 0.005 0.060 − 0.017313 − 303 3.225 3.224 − 0.001 0.047 − 0.010 − 3.236 − 3.244 − 0.008 − 0.056 0.002321 − 313 3.355 3.357 0.002 0.018 − 0.005 − 3.471414 − 404 3.426 3.428 0.002 0.074 − 0.016 − 3.449 − 3.460 − 0.012 − 0.082 0.007515 − 505 3.556 3.551 − 0.004 0.006 − 0.006 − 3.534 − 3.538 − 0.004 − 0.008 − 0.006110 − 000 3.596 3.594 − 0.002 0.032 − 0.009 − 3.621211 − 101 3.939 − 3.955 − 3.963 − 0.008 − 0.039 − 0.001514 − 413 3.978 − 3.997 − 3.991 0.006 0.098 − 0.017404 − 303 4.000 − 4.018 − 4.015 0.002 0.057 − 0.012312 − 202 4.129 4.130 0.002 0.041 − 0.011 − 4.140 − 4.149 − 0.010 − 0.047 0.002615 − 505 4.129 4.127 − 0.002 − 0.016 − 0.010 − 4.103 − 4.110 − 0.007 0.010 − 0.002415 − 505 4.138 4.136 − 0.002 − 0.046 − 0.010 − 4.114 − 4.120 − 0.006 0.043 − 0.003514 − 404 4.238 4.240 0.002 0.032 − 0.015 − 4.264 − 4.275 − 0.012 − 0.045 0.006413 − 303 4.279 4.284 0.006 0.075 − 0.019 − 4.264 − 4.277 − 0.014 − 0.083 0.010314 − 413 4.294 4.282 − 0.012 − 0.116 0.004 − 4.295 − 4.290 0.005 0.111 − 0.014314 − 404 4.540 4.544 0.005 0.024 − 0.018 − 4.561 − 4.574 − 0.013 − 0.033 0.008505 − 413 4.780 4.761 − 0.019 − 0.120 0.008 − 4.746 − 4.735 0.011 0.119 − 0.020213 − 303 4.811 4.813 0.003 0.006 − 0.001 − 4.843 − 4.854 − 0.011 − 0.009 − 0.008220 − 212 5.030 5.029 0.000 0.008 − 0.012 − 5.121 − 5.129 − 0.008 − 0.019 0.002220 − 303 5.048 5.053 0.005 0.082 − 0.020 − 5.127112 − 202 5.315 5.317 0.002 − 0.018 − 0.005 − 5.332 − 5.342 − 0.010 0.008 − 0.006011 − 101 6.002 6.004 0.002 − 0.027 − 0.005 − 6.020 − 6.030 − 0.009 0.018 − 0.008323 − 220 6.061 6.053 − 0.008 − 0.074 0.008 − 6.205 − 6.205 0.000 0.063 − 0.020222 − 112 6.246 6.242 − 0.004 0.013 − 0.009 − 6.343 − 6.346 − 0.002 − 0.019 − 0.004323 − 213 6.331 6.326 − 0.004 0.014 − 0.011 − 6.442 − 6.445 − 0.003 − 0.013 − 0.001121 − 011 6.503 6.499 − 0.004 0.002 − 0.013 − 6.605 − 6.607 − 0.002 − 0.009 − 0.001123 − 220 6.505 − 6.617 − 6.628 − 0.011 0.162 − 0.016221 − 211 6.586 6.581 − 0.005 0.014 − 0.005 − 6.696 − 6.701 − 0.004 − 0.019 − 0.005022 − 112 6.635 6.631 − 0.004 − 0.045 − 0.002 − 6.705 − 6.706 − 0.001 0.045 − 0.013322 − 312 6.697 6.691 − 0.006 0.003 − 0.004 − 6.802 − 6.805 − 0.003 − 0.007 − 0.006123 − 213 6.775 6.778 0.004 − 0.079 − 0.015 − 6.854 − 6.868 − 0.014 0.085 0.003423 − 413 6.799 6.790 − 0.010 − 0.032 0.003 − 6.877 − 6.875 0.001 0.027 − 0.012223 − 220 6.885 6.877 − 0.009 − 0.086 − 0.003 − 7.045 − 7.045 0.001 0.074 − 0.006323 − 413 6.910 6.902 − 0.008 − 0.060 0.007 − 6.975 − 6.974 0.000 0.056 − 0.018524 − 514 7.043 7.038 − 0.005 0.023 − 0.007 − 7.140 − 7.144 − 0.004 − 0.033 − 0.003423 − 404 7.043 7.052 0.009 0.110 − 0.017 − 7.140 − 7.159 − 0.019 − 0.119 0.007122 − 112 7.085 7.080 − 0.004 0.008 − 0.019 − 7.202 − 7.204 − 0.002 − 0.015 0.008223 − 213 7.155 7.150 − 0.005 0.002 − 0.022 − 7.282 − 7.284 − 0.002 − 0.002 0.013324 − 314 7.208 7.203 − 0.005 0.003 − 0.012 − 7.299 − 7.302 − 0.003 − 0.008 0.002222 − 312 7.440 7.435 − 0.005 − 0.045 − 0.003 − 7.533

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164315-13 Li et al. J. Chem. Phys. 139, 164315 (2013)

TABLE IV. (Continued.)

Obs., upper(J ′l′1l′ ← J ′′

l′′1 l′′ ) Obs., lower(J ′′l′′1 l′′ ← J ′

l′1l′ )

TransitionsDiff.(Calc.-Obs.) Diff.(Calc.-Obs.)

J ′l′1l′ − J ′′

l′′1 l′′ Obs. VMLRQ VMLRQ VMLR V12 Obs. VMLRQ VMLRQ VMLR V12

213 − 110 7.440 7.434 − 0.006 − 0.088 0.009 − 7.502221 − 111 7.510 7.507 − 0.004 0.008 − 0.003 − 7.628220 − 110 7.674 − 7.771 − 7.775 − 0.005 0.000 0.001322 − 212 7.721 − 7.834 − 7.839 − 0.006 − 0.007 − 0.004523 − 413 7.821 − 7.899 − 7.898 0.001 0.073 − 0.016423 − 313 7.823 − 7.930 − 7.935 − 0.006 − 0.001 − 0.004321 − 211 7.827 − 7.940 − 7.945 − 0.005 0.012 − 0.008524 − 414 7.853 − 7.952 − 7.956 − 0.004 0.009 − 0.004422 − 312 7.874 − 7.978 − 7.981 − 0.003 0.031 − 0.009323 − 313 7.937 7.935 − 0.002 − 0.032 0.000 − 8.028 − 8.034 − 0.006 0.028 − 0.010523 − 404 8.076 8.083 0.008 0.062 − 0.018 − 8.166 − 8.182 − 0.016 − 0.070 0.007324 − 414 8.320 8.317 − 0.003 − 0.047 − 0.013 − 8.413 − 8.418 − 0.005 0.042 0.004222 − 212 8.466 8.464 − 0.002 − 0.044 − 0.003 − 8.567223 − 313 8.762 8.759 − 0.003 − 0.045 − 0.012 − 8.868 − 8.874 − 0.005 0.039 0.004422 − 303 8.924 8.927 0.004 0.039 − 0.011 − 9.002 − 9.014 − 0.013 − 0.048 0.002122 − 212 9.305 9.302 − 0.002 − 0.049 − 0.013 − 9.419 − 9.425 − 0.006 0.043 0.003121 − 111 9.492 9.491 − 0.002 − 0.061 − 0.004 − 9.592 − 9.599 − 0.007 0.055 − 0.007331 − 321 10.038 10.035 − 0.004 0.027 − 0.006 − 10.319 − 10.324 − 0.005 − 0.032 − 0.003331 − 221 11.282 11.278 − 0.003 − 0.004 − 0.003 − 11.570

RMSD 0.007 0.056 0.010

with larger N. At N = 15, there is another distinctly changeto a reduced slope, indicating that the first solvation shellis formed, the filling of a second centered ring starts atN = 16. As shown in Figure 7, the calculated band originshifts on the current VMLRQ PESs are in remarkably goodagreement with experiment values34 from N = 17, which wasdetermined by subtracting the microwave transition frequen-

N

Δυ0

(cm

-1)

0 5 10 15 20-2.5

-2.0

-1.5

-1.0

-0.5

0.0

ExperimentVMLR

VMLRQ

V12

(para-H2)N-CO

10 12 14 16 18 20

-2.4

-2.2

-2.0

FIG. 7. Vibrational band shifts of CO in (para-H2)N clusters, obtained byPath integral Monte Carlo simulations using three different VMLR (open tri-angles), VMLRQ (open circles), and V12 (open squares) models for their dif-ference potentials, and compared with experimental values (filled circles).

cies from those of the a-type infrared transitions,25 assum-ing that the rotational frequencies are the same in the groundand first-excited vibrational states. On recent V12 potential,the calculated band origin shifts (open squares) as a func-tion of cluster size N in Figure 7, show a similar monotonicred decrease behaviour and slightly higher than those valuespredicted on the VMLRQ PESs, but with all the discrepanciesare smaller than 0.05 cm−1. The calculated band origin shifts(open triangles) on VMLR PES are almost the same values asthose obtained from VMLRQ PESs, because the correlation en-ergies �V

T(Q)int included in VMLRQ PESs are only calculated

on 4D intermolecular coordinates, which also indicating thatthe anisotropic potentials induced from the correction termof �V

T(Q)int have negligible effect on the band origin shifts,

although including it will improve the predicted infraredtransitions agree with experimental values in one order ofmagnitude.

V. CONCLUDING REMARKS

This paper presents accurate analytic vibrationally aver-aged 4D potential energy surfaces for H2–CO(v) complexesfor v = 0 and 1 which were obtained from five-dimensionalab initio potential energies. The PESs explicitly incorporatethe dependence of C–O stretching coordinate. The ab initiointeraction energies were obtained at the CCSD(T) level us-ing a complete basis set extrapolated from aug-cc-pVTZ andaug-cc-pVQZ basis sets, and with bond functions placed atthe midpoint on the intermolecular axis. In this work, electroncorrelation energies from the triple and quadruple excitations�V

T(Q)int are included, which calculated at CCSDT(Q) level

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164315-14 Li et al. J. Chem. Phys. 139, 164315 (2013)

using aug-cc-pVDZ basis set without bond function. The vi-brationally averaged potential energies were fitted to a 4Dgeneralization of the MLR potential form.54, 55 With and with-out the correction term of �V

T(Q)int , the fitted 4D potential

energy surfaces are denoted as VMLRQ and VMLR, respec-tively. The global 4D fit to the 30 206 interaction energies(<500 cm−1) had a root-mean-square residual of only 0.087and 0.082 cm−1 for v = 0 and v = 1 of VMLR potentials, and0.095 and 0.087 cm−1 of VMLRQ potentials, respectively, andrequired 196 fitting parameters. Comparison between VMLR

and VMLRQ potentials for v = 0 ground state of C–O, thedifferences at stationary points are anisotropic al, with 2.68,0.92, and 0.69 cm−1, for the global minimum, saddle point,and local minimum, respectively; while between our VMLRQ

and another recent high-accuracy V12 surfaces obtained from6D ab initio calculations, the differences at stationary pointsare significantly reduced to −0.77, 0.04, and 0.62 cm−1 .From this work, we also found that the potential energy differ-ences between vibrationally averaged ground v = 0 and firstexcited states v = 1 of H2–CO(v) along the minimum energypaths are almost the same, with energy difference smaller than0.02 cm−1.

Rovibrational energy levels for para H2–CO andortho H2–CO were obtained by the radial DVR/angular FBRmethod. For para H2–CO, total of 39 bound state energiesare yielded from our 4D-VMLRQ surfaces, which are in goodagreement with experimental values with the RMS discrep-ancies only 0.006 and 0.008 cm−1 for v = 0 or v = 1 states,respectively, tiny different from the values of 0.005 and0.007 cm−1 obtained from 4D-V12 surfaces. For orthoH2–CO, our 4D-VMLRQ surfaces support 103 intermolecu-lar rovibrational bound states, the energies relative to theasymptote limit 118.644 cm −1 of H2 with rotational excitedfor l2 = 1, are compared very well with those obtainedfrom 4D-V12 surfaces. The calculated band origin shift from4D-VMLRQ associated with the fundamental transition of COis −0.179 cm−1 for para H2–CO, results in same value fromthe experimental observation, and slightly better than thevalue of −0.176 cm−1 calculated from 4D-V12 surfaces. Thissuggests that our surfaces will yield reliable predictions forthe C–O vibrational shifts of CO in (H2)n clusters.

The calculated spectroscopic properties of our vibra-tionally averaged 4D VMLRQ PESs are in excellent agreementwith experiment: for 126 infrared transitions of para H2–CO,the RMS discrepancy is 0.007 cm−1, which is slightly bet-ter than that obtained from recent high accuracy surface ofV12. For ortho H2–CO, the RMS discrepancy for 179 boundstate transitions is 0.023 cm−1, which is slightly larger thanthat 0.018 cm−1 obtained from recent high accuracy surfaceof V12. For (para-H2)N–CO clusters, the calculated band ori-gin shifts as a function of cluster size N, show a monotonic reddecrease behaviour on the VMLRQ PES, which in good agree-ment with the experimental values, and also consistent withthose obtained from V12 PESs, with all the discrepancies overthe whole range from N = 1 to 20 are smaller than 0.05 cm−1.

As discussed above, our five-dimensional ab initio PESsfor CO–H2, that only explicitly incorporate the stretch coordi-nate of the CO and with rigid H2, can reach the same accuracyas recent high accuracy six-dimensional ab initio PESs. Inclu-

sion of the electron correlation energies from the triple andquadruple excitations �V

T(Q)int is the main contribution to im-

prove the accuracy of the potential energy surface. Other con-tributions such as those from the core-valence correlation cor-rections, larger basis set, and treatment of nonrigidity effectof H2, as incorporated in the V12 potential, have a very smalleffect on the predicted infrared transitions of the H2–CO com-plex. We also found that although the triple and quadruple ex-citations correlation energies �V

T(Q)int , can affect the infrared

transitions of the H2–CO complex, the band origin shifts forCO in (H2)n clusters are less sensitive.

ACKNOWLEDGMENTS

The authors thank Professor Piotr Jankowski (NicolausCopernicus University) for providing us with his V12 potentialfor the H2-CO complex. H.L. and X.-L.Z. were supported bythe National Natural Science Foundation of China (Grant No.21003058 and 21273094), by the Program for New CenturyExcellent Talents in University (H.L.). R.J.L.R. and P.-N.R.acknowledge the support given by the Natural Sciences andEngineering Research Council of Canada (NSERC).

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