Obtaining accurate pressure second virial coefficients for methane from an a b i n it i o pair potentialDavid H. Gay, Houfeng Dai, and Donald R. Beck Citation: The Journal of Chemical Physics 95, 9106 (1991); doi: 10.1063/1.461189 View online: http://dx.doi.org/10.1063/1.461189 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/95/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ab initio intermolecular potential energy surface and second pressure virial coefficients of methane J. Chem. Phys. 128, 214303 (2008); 10.1063/1.2932103 Methane-water cross second virial coefficient with quantum corrections from an ab initio potential J. Chem. Phys. 125, 014314 (2006); 10.1063/1.2207139 Potential energy surface and second virial coefficient of methane-water from ab initio calculations J. Chem. Phys. 123, 134311 (2005); 10.1063/1.2033667 A b i n i t i o potentials and pressure second virial coefficients for CH4–H2O and CH4–H2S J. Chem. Phys. 93, 7808 (1990); 10.1063/1.459362 Calculation of the second virial coefficients for water using a recent ’’a b i n i t i o’’ potential J. Chem. Phys. 64, 5308 (1976); 10.1063/1.432162

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Obtaining accurate pressure second virlal coefficients for methane from an ab initio pair potential

David H. Gay, Houfeng Dai, and Donald R. Beck Physics Department, Michigan Technological University, Houghton, Michigan 49931

(Received 4 February 1991; accepted 5 September 1991)

The pressure second virial coefficients, including fourth-order many-body effects, have been calculated for methane and are found to agree with the experiment on the average, to 2.8% over the temperature range 110-623 K using the basis set of Sadlej. This is a major improvement over the usual 30%-40% accuracy of ab initio potentials and also has been attained by us for H2 O. Monte Carlo simulations have also been performed with the potential and a C-C radial distribution function and the internal energy is obtained. The latter ( - 0.0757 eV Imolecule) is in good agreement with experiment ( - 0.0738 eV Imolecule).

I. INTRODUCTION

Recently, we have been obtaining dimer and some trimer intermolecular potentials for molecular species found in natural gas l

-3 for the purpose of obtaining thermophysi­

cal properties either by direct integration, as in the case of pressure second virial coefficients, or by Monte Carlo simu­lation for radial distribution functions, heat capacities, and chemical potentials. Since accuracies of 1 % or so are ulti­mately desired by the natural gas industry, this places rather severe constraints upon the potentials which we generate by ab initio means. 1-3 In fact, none of our potentials to date has attained this accuracy, nor does it appear that the potential for the H2 a dimer, also of interest to the natural gas indus­try, has achieved this accuracy (for some recent work see Refs. 4 and 5, and references therein).

On the other hand, there has been some recent work by Szczesniak et al.6 on the CH4 potential (principally for dimers) which, because it seems to represent a significant improvement over previously published ab initio potentials, deserves to be explored to determine what kinds of accura­cies it is capable of producing for thermophysical properties.

II. GENERATING A SECOND-ORDER AS INITIO INTERMOLECULAR PAIR POTENTIAL

Throughout this work we shall be generating the meth­ane intermolecular pair potentialby assuming the validity of the Born-Oppenheimer approximation and maintaining the individual methane molecules as rigid entities whose angles and distances are fixed at the experimental values.

We begin by selecting a point on the intermolecular sur~ face that we wish to explore and obtain the potential energy by doing an ab initio calculation for which the zeroth-order approximation is generated by doing a Hartree-Fock calcu­lation for the closed-shell pair of methane molecules. For nonpolar species such as methane, by far the greatest contri- . bution to the attractive part of the potential occurs from dispersion effects, for which the largest term is proportional to the product of the dipole polarizabilities and inversely proportional to the sixth power of the internuclear separa­tion (London equation). Much weaker attractive contribu-

tions arise from the presence of octupole and higher mo­ments which give rise to electrostatic and induction effects.7

Since it is well known that the Hartree-Fock approximation does not account for any dispersion effects, these results are unacceptable. We improve the Hartree-Fock result by em­ploying many-body perturbation theory8 (MBPT) which is classified according to its order: second (E2), third (E3), fourth (E 4), etc. Since second-order results only involve the first nonzero contribution to the dispersion energy, a priori it is doubtful whether they can suffice, particularly where high accuracy is desired (as is the case here). Furthermore, both third- and fourth-order effects have been found4.6 to be of similar importance, which should not be too surprising as both arise, in part, from the second-order wave function.

While all of E 3 and a portion of E 4 can be computed at relatively little extra cost, the triple excitation8

•9 portion of

E 4 is computationally expensive and also represents a sub­stantial portion of the total E 3 + E 4 correction. In Sec. IV we give the details of how we obtained theE 3 + E 4contribu­tion.

The geometries we choose to explore are the six orginal­ly chosen by Kolos et aI., 10 labeled A-F, in their Hartree­Fock (HF) study of the methane pair potential. These geo­metries represent both the most attractive (A) as well as more repulsive (E and F) portions of the potential surface. Figures displaying the geometries are available in the work of Kolos et al. 10 and also that of Szczesniak et al.6

In doing ab initio molecular calculations, the choice of basis set (one-electron contracted Gaussian functions cen­tered on different atoms in the molecule) can be crucial. Sadlejll has recently put forward a new basis set of modest size which he carefully checked to make sure that it could account quite well for the lower-order permanent moments and polarizabilities of individual molecules. This set l1 con­sists of26 functions for each C atom present and 9 functions for each H atom present. Thus a total of 124 functions is needed for each ab initio point generated for the potential surface. For the HF and E 2 portions of the calculation, ap­proximately 11 CPU hours are needed on the SUN SP ARC2 per point. With the 126 function basis, calculation times for E 3 + E 4 would be about 2000 h per point, which was too

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Gay, Dai, and Beck: Second Yirial coefficients for methane 9107

much to enable the full surface to be calculated with this basis, so an alternative approach was taken (see Sec. IV).

In doing ab initio calculations for the potential energy, large errors in the directioIlI of too large binding energies can be made if basis-set saturation errors (BSSE) are not ac­counted for. These arise diue to the incompleteness of the basis set used to describe the individual molecules and so are enhanced for smaller basis sets. What happens is that the basis set associated with molecule A contributes both to the intermolecular potential energy (which we desire to calcu­late) and lowering the energy of molecule B. If this latter term is not removed, then it is falsely assigned to the inter­molecular potential energy. Boys and Bernardi 12 suggested how this might be removedl to a large extent. In the counter­poise method 12 the energy of each monomer varies for each point on the potential surf.'lce due to the changing effective­ness of the second molecule's basis set in lowering the energy of the first monomer. The counterpoised monomer's energy is evaluated by removing the electrons and nuclear charges from the second molecule, but not the basis functions. This can nearly triple the calculation times. Some savings occur because one does not need to recalculate the two-electron integrals, and in cases where due to symmetry, the counter­poised energy of the two monomers is identical. The inter­molecular potential energy for a point on the surface then becomes the difference of the dimer's total energy and the sum of the two counterpoised monomer's energy. The accu­racy of the method has been tested4,Io by seeing how the potential energy varies with basis-set size; ideally (Le., larger basis sets) it should be almost constant. For our purposes, this seems to be the case.

Using the Sadlej II basis set, Szczesniak et al.6 obtained HF + E 2 results for four of the six geometries of Kolos et al.,l0 viz. A, E, D, and F. This represents an insufficient num­ber of points to establish the potential surface. In this work, we provide in Table I points in the C and E ge()metries, as well as some additional points for geometries A, E, D, and F. Unlike our earlier work,I-3 we have generated the results using the GAMES computer code. 13 This choice was made because the integrals code was faster by about 4-5 ti.rp.es than our earlier code l - 3 and also produced a (smaller) speed gain at the E 2 MBPT level. On the other hand, disk usage in­creased (about twice as much was needed) and this will be significant when larger molecules are studied.

III. FITTING THE SECOND-ORDER (£2) SURFACE

To date, fitting the ab initio points (second-order, E2 surface) still remains something of an art form. Among oth­er considerations are (1) how many ab initio points to use; (2) what potential form to use; (3) in what, sequence the potential parameters are to be determined; and (4) are all parameters to be determined by least squares or are some to be determined by matching physical properties such as per­manent moments?

In methane we have one of the simpler species: it lacks low-order permanent mOn1~ents (the octupole moment is the lowest-order surviving moment) and is "nearly" spherical. Thus we might expect to be able to make good use of accu-

TABLE I. Additional points added to Szczesniak's Table V' for six orienta-tions of (CH4 la in /LH.

Configuration R (a.u.) E (SCF) E2 Total

A 6.80 857.1 - 1364.2 - 507.1 A 7.20 405.6 -994.7 - 589.1 A 11.0 0.30 -75.89 -75.59 A 15.0 0.12 - 11.19 - 11.07 A 16.0 0.08 -7.46 -7.37

B 6.80 1418.2 - 1551.1 -132.9 B 7.20 685.4 - 1122.0 -436:6 B 11.0 1.37 -80.50 -79.13 B 15.0 -0.05 - 11.42 - 11.47 B 16.0 -0.04 -7.49 -7.53

C 7.00 1380.0 - 1419.1 - 39.1 C 7.50 557.0 - 953.4 - 396.4 C 8.5 80.18 444.91 - 364.73 C 9.00 27.0 323.2 - 296.21 C 10.0 0.72 156.88 - 156.16 C 11.0 - 1.76 - 85.49 - 87.25 C 15.0 -0.54 - 11.41 - 11.95 C 16.0 -0.35 - 8.01 -8.3

D 11.0 -6.09 - 91.97 -98.06 D 15.0 -1.42 - 12.72 - 14.13 D 16.0 -0.92 -8.46 -9.38

E 7.50 1332.1 - 1289.1 --,-43.0 E 8.50 196.0 -574 - 378.8 E 9.00 67.1 -430.9 - 363.8 E 10.0 3.1 -197.6 - 194.5 E 11.0 - 3.6 -98.7 ~ -102.5 E 12.0 -2.64 - 54.84 - 57.48 E 15.0 , -0.65 -12.61 - 13.27 E 16.0 -0.38 - 8.56 - 8.91

F 7.70 2914.02 - 1655.75 1258.27 F 7.90 2076.43 - 1377.53 698.90 F 11.0 18.02 - 122.15 - 104.13 F 11.5 10.97 - 87.06 -76.09 F 12.0 7.12 - 64.12 -57.0 F 15.0 0.9 -14.28 -13.38 F 16.0 0.3 -9.36 - 9.04

"Reference 6. The "total" value at R = 7.0 a.u., geometry A, should be - 575 instead of - 523.6/LH.

rate studies of fits made for the rare gases. In this context we have in mind the studies of Tang and Toenniesl4 on Ar mix­tures. Their results suggest using a combination of a damped attractive term and an exponential repulsive term on the heavy atom (e).

We begin our fit with the purely electrostatic part of the potential, on the general principal that such effects can be large. To account for the lowest-order effect, the interaction of the octupole moments, we place a + 4a.u. point charge on each C atom and a-I a. u. charge on each H atom. The absolute values of the charges are then adjusted to match the Hartree-Fock value, 2.1629 a. u., for the octapole moment.6

Since the octapole moment is given by the formula (e.g., Ref. 6):

5 5 n. = - L q,(x,y,z/i(r, - d/»·

2 1=1

For methane, this reduces to IOd 3, where d is the cube edge

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9108 Gay, Dai, and Beck: Second virfal coefficients for methane

associated with the methane molecule (C at the center, H's at the vertices of the cube), i.e., d = 1.183 568 a. u. This means each change must be multiplied by 0.1304538 a.u. to yield the Hartree-Fock octapole moment. The interaction constant for the octapole-octapole interaction, shown in Ta­ble II, is just the square of this value, converted to eV (1 a.u. = 27.209 76 eV) and it remains fixed throughout the fitting process. We have chosen to fit the Hartree-Fock, rather than the experimental, value of the permanent mo­ment because we wish to fit the ab initio results as well as possible rather than trying to correct them during the fitting process.

We have examined the contribution from the octapole interaction by direct evaluation of the "point-charge" expression in Table II for the A geometry. For large R (R> 10) it varies nearly as R --7, its theoretical limiting val­ue, and is about 4% of the total potential (see Table III). For the smallest R, 6.5 a. u., it is about 23 % of the total. While a damping factor has been included (see Table II), this is inef­fective for R > 6, but somewhat less so for the E 4 potential. We may also note that use of the point-charge expression of Table II, while quite simple, involves some cancellation dur­ing evaluation (five digits are lost, so calculations must be done in double precision). This expression, of course, con­tains contributions from higher multipoles, such as the hexa­decapole, but these are both small and unreliable (see the Discussion section).

Next, we focus on the "tail" region of the potential, on the basis that we wish to work from the simplest to the more complicated (well region) part of the potential. Twelve

TABLE II. Parameters of the analytic CH .. dimer potential fit to the ab initio points (distance in a. u., energies in e V).

C-C interaction term

b ~ C2n VCR) = A exp( _. R) - £.J---'; n;.a.3 R

Coefficient Second order Fourth order

b 2.880000 ODD + 00 2.765800 OOD + 00 A 0:922 500 OOD + 07 0.71610000D +07 C6 0.393 300 OOD + 04 . 0.409 423 02D + 04

C. 0.360 100 ODD + 05 0.347900 OOD + 05 CIO - 0.236 000 ODD + 07 -0.566 900 OOD + 06

CCH Coulombic interaction term

[ 1 ~ 4 16 ] V(R)=A L-- I-+ I-·- [1-exp(-bR)]

H-H R C-H R c-c R

Coefficient Second order Fourth order

b 0.526 500 OOD + 00 0.299 600 ODD + 00 A 0.463 100 OOD + 00 0.463 100 OOD + 00

H-H exponential term: VCR) = Al exp( - bl R)

Coefficient Second order Fourth order

At 0.127 50000D + 03 0.335 200 DOD + 09 bL 2.120 000 ODD + 00 6.189900 OOD + 00

points, two in each of the six geometries, are used (R = 15 and 16 a. u.), and a C-C 1/ R 6 interaction term is introduced. At this stage, the fitting error was 8.3%.

In the next stage, seven more points (R = 10-12) were introduced and a C-C 1/R 8 interaction term was fit (fitting parameters from previous cycles are frozen throughout). The average error is now 11.7%. In the third stage, eight more points (R> 8.5 a.u.) are brought in and a 1/R 10 term is fit, with an average error of 11.9%.

Finally, every point (a total of 50) with energy < kT (T = room temperature) is included and the nonlinear pa­rameters are fit. This includes exponential replusive terms for H-H and C-C interactions and a damping factor for the octapole-octapole interaction (see Table II). For this final fit, the average error is 24.5% with a standard deviation of 0.0012 eV (7% of the well depth for the E2 surface).

Many other types of terms were tried and found wanting at various stages of the fitting process. These include the following: (1) damping the C-C attractive terms (accom­plished now by restricting the use of the potentials of Table II toR> 6.5 a.u.), (2) having an attractive term for the H-H interaction, and (3) allowing both attractive and replusive terms for C-H interactions.

The potentials of Table II have the quality of being sim­ple enough to be easily used in simulation and other studies. The octapole damping factor could be removed to further simplify the potential.

IV. OBTAINING AND FITTING THE £4 SURFACE

Different orders of many-body perturbation theory are conveniently described in terms of "diagrams" which are reduced to sums over products of two-electron integrals di­vided by products of differences of orbital energies obtained from the Hartree-Fock process.8 There is one such diagram in second-order (E2) with four summation indices, three diagrams in third order (E 3) with six summation indices and 39 diagrams in fourth order (E 4) with eight summation indices (for E 4, a few of the diagrams may be combined). In the case of closed shells, it is computationally convenient to replace the sums over spin orbitals with sums over orbitals. 15

Sums are then either over occupied orbitals (5 for the CH4

monomer, or 10 for the dimer) or unoccupied orbitals (e.g., 116 = 126 - 10 for the dimer). Evaluation of theE 2 and E 3 contributions to the potential energy is computationally rou­tine. As part of this process, the original atomic orbital (AO) two-electron integrals are transformed to molecular­orbital form using the Hartree-Fock solutions. There are approximately 31 million of these.

Certain ofthe fourth-order diagrams currently present a computational bottleneck. For each term in the eightfold sums there are five multiplications and one division that have to be done and there can be as many as 1014 terms that need evaluation. However, for all but a small number of E 4 diagrams, called the triples,8 the amount of calculation can be dramatically reduced (e.g., Ref. 9) so that computation costs are comparable with those for evaluation of E 3. The triples cannot be neglected. In the aggregate they provide

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Gay, Dai, and Beck: Second virial coefficients for methane 9109

TABLE III. Energy contributions to methane dimer binding energy using diffuse basis set (in hartrees).

Geometry" UHF 82 E3 84 Total

A 6.5 0.001618 -0.001108 0.000016 - 0.000078 0.000448· A 7.0 0.000640 - 0.000777 0.000001 -0.000056 -0.000192 A 7.5 0.000216 -0.000555 -0.000003 - 0.000061 -0.000403 A 7.75 0.000 106 - 0.000474 - 0.000 002 - 0.000036 -0.000406 A 8.0 0.000037 -0.000404 -0.000004 -0.000031 - 0.000402 A 8.5 -0.000028 -0.000 299 -0.000002 - 0.000 021 -0.000 350 A 9.0 -0.000047 -0.000224 - 0.000001 -0.000019 -0.000291 A 10.0 -0.000036 - 0.000 128 -0.000002 -0.000012 - 0.000178 A 11.0 -0.000020 -0.000074 - 0.000001 - 0.000009 -0.000104 A 12.0 -0.000009· - 0.000045 - 0.000001 - 0.000006 -0.000061

B 7.5 0.000 391 - 0.000 595 - 0.000 010 -0.000042 - 0.000256 B 8.0 0.000 110 - 0.000431 - 0.000008 - 0.000033 -0.000362 B 8.5 -- 0.000 013 - 0.000 312· -0.000006 - 0.000024 - 0.000 355 B 9.0 -0.000037 - 0.000235 - 0.000005 - 0.000018 -0.000295 BlO.0 ~- 0.000037 - 0.000 134 - 0.000003 - 0.000012 - o.ono 186 B 11.0 -;- 0.000020 -0.000078 - 0.000003 - 0.000009 - 0.000110 B 12.0 -0.000011 - 0.000 047 - 0.000 001 -0.000005 -0.000064

C 6.0 0.008562 - 0.001764 0.000016 - 0.000 107 0.006707 C7.0 0.001585 -0.000 846 - 0.000 017 - 0.000053 0.000669 C 7.275 0.000970 -0.000705 - 0.000016 -0.000075 0.000174 C 7.4 0.000771 - q.000648 - 0.000014 - 0.000043 0.000066 C 8.0 0.000230 -0.000440 - 0.000 012 - 0.000030 - 0.000 252 C 8.4 0.000080 - 0.000 344 - 0.000009 -0.000026 -0.000299 C 9.0 - 0.000015 - 0.000 242 -0.000007 -0.000019 -0.000283 ClO.0 - 0.000036 - 0.000 137 - 0.000003 - 0.000013 - 0.000189 ClI.O ~~ 0.000 023 - 0.000082 - 0.000 001 - 0.000008 - 0.000114 C12.0 -'- 0.000 010 - 0.000050 - 0.000001 - 0.000008 -0.000069

D 7.4 0.001224 - 0.000 773 - 0.000008 - 0.000039 0.000404 D 8.0 0.000 353 - 0.000 512 - 0.000 012 - 0.000030 - 0.000 201 D 9.0 --0.000003 - 0.000273 -0.000006 - 0.000018 -0.000 300 D 10.0 -- 0.000038 -0.000153 -0.000004 - 0.000012 - 0.000 207 D 11.0 - 0.000029 -0.000090 - 0.000003 -0.000009 - 0.000 131 D12.0 -- 0.000 019 -0.000054 - 0.000001 -0.000006 -0.000080·

E 7.0 0.004172 -0.001074 - 0.000017 - 0.000047 0.003034 E 7.5 0.001728 - 0.000736 - 0.000025 - 0.000036 0.000931 E 8.0 0.000 663 -0.000523 - 0.000022 - 0.000027 0.000 091 E 9.0 0.000043 - 0.000280 - 0.000 013 - 0.000019 - 0.000269 EI0.0 .- 0.000038 - 0.000 159 - 0.000006 - 0.000 014 -0.000217 E 11.0 - 0.000030 - 0.000093 -0.000004 - 0.000008 - 0.000.135 E12.0 .- 0.000017 - 0.000057 - 0.000002 -0.000005 -0.000081

F 6.5 0.022877 - 0.003 481 0.000 376 - 0.000114 0.019658 F 7.0 0.009989 - 0.001953· 0.000104 - 0.000049 0.008091 F 7.5 0.004306 - 0.001157 0.000002 -0.000030 0.003 121 F 8.0 0.001773 -0.000736 - 0.000027 -0.000023 0.000987 F 8.6 0.000 540 - 0.000466 - 0.000028 - 0.000021 0.000025 F 9.0 0.000206 -0.000355 - 0.000023 -0.000019 - 0.000 191 FlO.0 -0.000 032 - 0.000192 -0.000012 - 0.000015 - 0.000 251 Fl1.0 -0.000046 - 0.000110 - 0.000007 - 0.000009 - 0.000172 F12.0 - 0.000029 -0.000067 -0.000004 - 0.000006 - 0.000106

A' 6.0 0.004227 - 0.001656 0.000039 -0.000112 0.002498 A' 6.6 0.001421 - 0.001 063 0.000004 -0.000074 0.000288 A' 9.0 - 0.000050 - 0.000228 - 0.000002 - 0.000020 -0.000 300 .1'10.0 - 0.000011 -0.000046 - 0.000001 - 0.000006 -0.000064

B' 7.0 0.001068 - 0.000 834 - 0.000010 - 0.000056 0.000 168 B' 7.4 0.000482 - 0.000 635 - 0.000010 0.003062 0.002899 B' 8.0 0.000 109 -0.000431 - 0.000008 - 0.000033 - 0.000 363 B' 8.6 - 0.000013 - 0.000 297 - 0.000006 -0.000024 - 0.000 340 B' 9.0 -=-0.000038 - 0.000235 - 0.000005 -0.000018 - 0.000296

• Geometries A-F have been standardized earlier (Ref. 10). Geometry A ' differs from A in that the H atoms of one molecule are rotated around the C-C axis so that one has an eclipsed configuration. B ' differs from Bby a 90' rotation about the C-C axis. The number in this column indicates the C-C distance in a. u.

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9110 Gay, Oaf; and Beck: Second vfrlal coefficients for methane

150% of the total E 4 contribution at the minimum (A geom­etry, R =7.75 a.u.).

We have recently completed the development of a closed-shell E 4 code wliose first application is reported here. We have tested it by comparison with the published results of Krishnan and Pople9 for the Ne atom and the agreement is excelIent Two routes wen~ 9pen to us to attempt to pro­vide the large-scale computing capacity needed to treat the full basis set (126 functions). The first would have been to get sufficient access to a supercomputer. Probably > 200 hours of time would be needed (see also the comments in Ref. 4). The alternative was to use a parallel processing ap­proach. We chose the latter and constructed a "cube': of five Sparc 1 + computers. We "spread" our E 4 code over these by the simple expedient of breaking up the outer summation loop so that each of the parallel cpus did 20% of the calcula­tion. Since there was no intercommunication, little perfor­mance degradation was observed (Le., we had the power of all five Sparc 1 + 's). Even with this enhancement, we esti­mate that it would have taken about 400 h of "cube" time per point, so some further simplification seemed necessary.

We thus sought a smaller basis set. Our recent experi­ence16 in obtaining accurate polarizabilities of alkanes using very minimal sets (33 functions for a methane molecule) suggested this set as a possible one. We report outJlb initio results for this set in Table III. Although the E 2 results are significantly higher than those obtained with the .. set of Szczesniak et af. 6 we are really seeking to establish the per­centage correction that E 4 provides the E 2 surface. A com­parison of the single E 4 value obtained using the larger basis (at the dimer minimum) to their6 E2 value yields substan­tially the same ratio as our small (or "diffuse" 16 ) basis-set ratio at the minimum.

The E 3 + E 4 corrections are then added to the E 2 sur­face as a percentage correction (determined from the small set) of the la,rge set's E 2 values. The ab initio po}nts are refit with the parameter values shown in Table II. The average error of the fit is 26.8% and the standard deviation is 0.00163eV.

From Table III it can be seen that the E 4 corrections provide about 10% of the potential energy for the attractive regions of the surface, whereas the E 3 contribution is several times smaller. Within the E 4 contribution, the triple excita­tions contribute 150% of the total.

It would be valuable to be able to physically interpret the E 3 + E 4 contributions to the energy surface. The dispersion energy is proportional to the product of the polarizahilities, a(m)acn ), where m and n indicate the order, in the perturba­tive sense, to which the polarizability is known. Potential­energy results of order N eN = 2,3,4 here) then involve all polarizability orders through m + n + 2 = N. This means the E2 energies involve zeroth-order polarizabilities only. Let us consider the dipole polariiabilities only e they are the largest). Using the expression in oUf earlier work;r6 we have calculated this quantity for the Sadlejll basis, obtaining a zeroth-order value of 13.17 a.u. We find it interesting to note that this value differs very little from the one we obtained earlierll using a much smaller basis set. This is consistent with the upper-bound property16 associated with a(O) (the

smaller basis set was developed by maximizing a(O». To further support the utility of the small basis set,16 we note that the small set coupled Hartree-Fock (CHF) value ob­tained for a was 18.10 a.u. compared with the 16.01 of the Sadlejll set (the experimental value17 is 17.23 a.u.).

For N = 4 (E 4), dispersion terms involving a (2) ap­

pear. It would be useful to have second-order polarizabilities for this molecule and in general, but they are seldom avail­able. We do have availablell a coupled Hartree-F<1ck value including many-body effects which in principle gives a to all orders; the value is 16.59 a.u. This would indicate that E2 contains only about 63% of the dispersion energy and it is not unreasonable to suppose the E 3 + E 4 contains a good part of the remainder. However, the overall correction from E 3 + E 4 to the potential surface is on the order of 10%, not 37%. Another correction can arise from the ionization po­tential (IP) on which the dispersion energy depends linear­ally on the London formula for two identical molecules. The Sadlej basis yields 0.5246 e V for the IP through second order (the Hartree-Fock value is 0.4891 eV), whereas the experi­mental value18 is 0.4774 eV. This suggests that the disper­sion energy might be reduced 10% from this cause when the IF is fully corrected. There are also corrections to higher­order dispersion energy terms as well. For example, the 1/ E 8 terms depend on the quadrupole polarizability of the molecule. We have computed this at the zeroth-order level and find thexx,YY, ZZ, xy, XZ, andyzcomponents to be 337.6, 337.6, 1138.0, 66.0, 666.0, and 666.0 a.u. Our studies! in H 2 S suggest that without! functions on the "large" atom, these values could be as much as 50% too low. Since this contribution is an attractive one, the dispersion energy would be increased from this cause.

V. SECOND VI RIAL COEFFICIENTS, 8fT)

The second virial coefficients are obtained by direct evaluation of the six-dimensional integral using a modified version of Murad's 19 computer program. The integral is di­vided into four parts, basedoii the C-C distance R. For the first region R < 0.9s (s = 5.588 a.u.) a hard-core potential is assumed which contributes (2rr/3)N.4 (0.9S)3 = 23.774 cm3/mol to the classical second virial coefficient, B(T), which is given by the expres.sion

B( T) = - NA J R 2 dR (11" dB sin 0 (211" drp (11" d02 16r Jo Jo Jo

. (211" (211" _ X sin B2 Jo drp2 Jo dX2 (e-f:lV(R;a) - 1),

where R is the vector joining the center of masses of molecule one and molecule two, R = IR I, O,rp is orientation of R, (rp2,B2,X2) is the orientation (Euler angles) of molecule

_. two. V(R;a) is the pair potential, where a is the orientation of the two molecules, /3= l/kT, and N.4 =Avagaqro's number.

In the second 0.9s<R <2.0s and third regions 2.0s < R < 8s the integral is evaluated numerically using Stroud's method.20 In the outer region, the radial mesh is coarser because the potential is smaller and more isotropic

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Gay, Dai, and Beck: Secondviriai coefficients for methane 9111

there. After some trial and error, a mesh of 99 radial points and five angular points (each dimension and region) was selected. The test used was that B(T) must be stable to 1 cm3/mol or better for low T and to 1 % or better for high T. Evaluation of B( T) for this mesh took - 9 computer hours on a Sparc 2. We may note that our mesh is approximately twice as fine in each dimension as. that used by Murad in his sample data see9 for CH4 •

Finally, we approximately evaluated the integral in the range 8s < R < 00 by arguing that (1) the potential was small, so the exponential could be expanded and only the first term kept, i.e., integrand = - V IkT;(2) that for these values of R, V was isotropic; and (3) that V varied as 1/ R 6.

From this it follows that an integral in the range (RL,R U) contributes A [(RL -3 ~ (RU) -3] toB(T). We evaluated the constant A by choosing it to fit the value obtained from the Stroud method for the last four intervals in the third region. Four intervals were found to adequately smooth the tail behavior. Then a tail correction was obtained from the expression A I(RL )3.,ThestabiIity of this process was tested by changing the outermbs.t R (8s) and reevaluating the tail contribution. These are always negative and largest (in mag-

nitude) for the lowest T( - - 0.7 cm3/mol and dropping to - - 0.1 cm3/molat T-600 K). B(T) was stable when this was done.

We also made a plot (histogram) of the B( T) integrand . as a function of R to determine which were the important regions. We chose as the well region 7 <R <9.5 a.u. and fomid that the tail and well regions were equally important for all T and that the repUlsive (R < 7 a. u.) region was of similar importance for high T, but less so for low T.

We also obtained quantum corrections to B(T) using the formula in Hirschfelder, Curtiss, and Bird21 for a 6-12 Lennard-Jones potential. We took the potential parameters from the work of Murad et aJ.22 It has been pointed out that T Pack23 has corrected the formula of Hirschfelder, Curtis, and Bird. Evidence23 suggests that these corrections can be significant for substanti~I.lly anisotropic potentials, or for very light molecules. Since neither condition is met in this case, and additionally the quantum corrections obtained are already small, we did not make further use of the T Pack results.

The results are shown in Table IV and Fig. 1. The E2 ( + QC = quantum corrected) result has an average abso-

TABLE IV. Second virial coeffi<:ient, BeT), fo.r CH4 including forirth-order effects 'and quantum corrections (in cm3jmol).

Temperature . .

-

(K) E2" E2+QCb E4+QC' Expt.d "",Errore

110,00 - 309.10 - 305.47 - 336,22 - 330 1.9 120,00 . - 261.01 - 258.17 - 283.21 -273 3:7 130,00 -224,05 - 221.76 - 242,60 - 235 , 3,2 ...... -140.00 -194.83 - 192.93 - 210.56 -207 1.7 150.00 - 171.16 - 169.55 - 184.48 -182 1.4 160,00 - 151.62 - 150,24 - 163,38 -161 1.5 . 180.00 - 121.28 - 120.22 - 130,40 -129 1.1 200,00 - 98.86 - 98.02 - 106.09 - 105 .. 1.0 210.00 - 89.71 - 88.95 -96,20 - 95,30 0,9 220.00 - 81.62 - 80,92 '- 87.48 - 86,60 1.0 230.00 -74.43 -73.79 .... _·79,69 -78.90 ·1.0 234.05 - 71.74· -71.12 -76.78. - 75.90 1.2 240.00 - 67.99 - 67.40 -72.74 -72.00 1.0 260.00 - 56.94 - 56.44 -60,84 - 60.20 1.1 263.08 - 55.42 - 54.93 - 59.20 - 58.35 1.5 273,00 - 50.75 - 50,29 - 54.47 - 53,35 2,1 298.15 -40.79 -40.40 - 43.48 -42,82 1.5 323.15 - 32.67 - 32.33 - 34.76 - 34.23 1.6 348.15 . - 25.92 - 25.62 - 27.53 - 27.06 1.7 373.15 - 20,23 -19.96 - 21.43 - 21.00 2,1 398.15 - 15,38 - 15.14 - 16.24 - 15.87 2,3 423.15 - 11.18 - 10.96 -11.75 - 11.40 3.1 448.15 -7,53 -7,33 -7,85 -7,56 3,8 473.15 -4,32 ; -4.14 -4.42 - 4.16 6.3 498.15 -1.48 '- 1.13 -1.39 - 1;16 19.8 523.15 1.05 1.21 1.32 ,1.49 - 11.4 548.15 3.32 3,47 3.73 3.89 -4,1 573.15 5.36 5.49 5.90 5.98 -1.3 598.15 7.20 7.33 7.87 7.88 -0,1 623.15 8,88 9.00 9.65 9,66 . -' 0.1

• Second-order resuli;using Szczesniak et at. (Ref. 6) basis. b a + quantum corrections (see text). C b + faurth-order correctians abtained using the diffuse basis set. dFram Dymand and Smith (Ref, 24). ·Percentage errar from calumI1s 4 and 5.

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9112 Gay, Dai, and Beck: Second vi rial coefficients for methane

0.0

-100.0

8(T) (cm3/mole)

-200.0

-300.0

100 200 300

E2+QC

E4+QC

• Experiment

400 500

Temperature (K)

FIG. 1. Second virial coefficients B( n for methane in cml/mol.

600

lute error of 6.3%, and the E 4 + QC result has an average absolute error of 2.8% when compared with the experimen­tal results of Dymond and Smith24 (see also Ref. 25). The rms errors are 7.71 and 2.80 cm3/mol, respectively. This can be best appreciated by comparing the results obtained (here) from a standard exp - 6 potentiaf2 which yields average absolute errors of -45%

VI. MONTE CARLO SIMULATIONS

To provide another test of the potential, Monte Carlo simulations were made for a dense liquid state. The tempera­ture chosen was 130 K with a density of 40.16 cm3/mol. These conditions were selected to allow comparison to ear­lier simulation studies.22 Constant NVT simulations were done with 500 and 834 molecules. For the E2 potential a single simulation was done with 500 molecules. Two simula­tions were done with the E 4 potential, one using 500 mole­cules and another with 834. The second run was used to estimate the long-range correction to the excess internal en­ergy «0.00015 eV/molecule). Another simulation using 256 molecules and the. William's22 semiempirical site-site potential for methane was done to benchmark our Monte Carlo program and provide a reference radial distribution function. The results of this run agreed with the earlier pub­lished molecular-dynamics results of Murad et al.22

All our Monte Carlo simulations used a face-centered­cubic lattice for the initial state. The duration of the equili­briation phase of the simulation was determined by the aver­age root-mean-squared (rms) displacement of the molecules. A displacement of 6.5 A was used to determine when the lattice had "melted," which should be adequate since this is quite a bit larger than 0.5uwhich is - 3.2 A. This resulted in 98 K configurations in the simulation using the E 2 potential and 64 K for theE 4 potential. Both simulations used 1000 K configurations sampled every 2 K configura­tions to obtain t4e radial distribution function and excess

justed so that the acceptance rate was - 50%. CPU times were ~one day for each run on the SPARC 2.

The three radial distribution functions are shown in Fig. 2. The biggest difference in the three distribution func­tions is the height of the first peak, which is ,....;3.0 for the E 4 curve and 2.5-2.6 for the others. This is because the potential is deepest for the E 4 potential well. There is no experimental data available which we can compare to.

The excess internal energies have been calculated, but the long-range corrections have been neglected. The long­range corrections should be small, and this is indicated by the 834 molecule simulation with the E 4 potential (the strongest of the three and therefore the one that would have the largest long-range corrections). The computed excess internal energies in eV Imolecule are - 0.0582, - 0.0757, and - 0.0679 for the E 2, E 4, and MuradlWilliams poten­tial, respectively, while the experimental value is - 0.0738 e V Imolecule.22 The rms variation in the internal energy was 0.0005 eV Imolecule for all of the runs and the two E 4 results agreed within this error. For the internal energy the agree­ment with experiment is good for the E 4 potential (within 5%); this is normally the most accurate quantity calculated with Metropolis Monte Carlo type simulations.

VII. DISCUSSION

This work has demonstrated that it is possible, at least for some pair potentials, to reduce ab initio errors in B( T) from an unacceptable 30%-40% to 2%-3% iffourth-order effects are included in the surface. Using only second-order MBPT, errors are -5%: We have seen that evaluation of the triples portion of E 4 is computationally quite intensive. It obviously would have been preferable to make these compu­tations with the original basis set, and from a computer hard­ware perspective this seems to melin either having access to a

3.0

2.0 - E4 Potential

--- E2 Potential

g(r) ...... Murad

1.0

0.0

0.0 10.0 20.0 30.0

C-C Separation (a.u.)

FIG. 2. C-C radial distribution function for methane.

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Gay, Dai, and Beck: Second virial coefficients for methane 9113

significant block of supercomputer time, or attaching a "dedicated" vector processor to a powerful workstation. We believe the second choice to be the more cost effective one. Undoubtedly, further progress in the future will be made in developing even smaller basis sets which retain essentially the same accuracy of the Sadlej set. II Another approach which was tried during the course of this work was to ex­clude "virtuals" (unoccupied Hartree-Fock orbitals) which did not contribute significantly to the zeroth-order dipole polarizability (the triples time scales as the fourth power of the number of virtuals). However, comparison of the two runs (full and partial set ofvirtuals) showed an unaccepta­ble change in the potential (the energy of the counterpoised monomer changed most dramatically), so this approach was not explored further.

From the results of the earlier sections, it should be clear that it is difficult to obtain a fitted potential that behaves according to one's "intuition." In part this arises because the "intuitive" forms, such as those for large R (see, e.g., Refs. 7 and 26), work best for R > 20 a.u., a region not considered here (questions of numerical accuracy might arise then). Due to a large extent to the nonlinear parameters used, there is not a unique fit, but families of fits, all more or less equally acceptable based on a least-squares test. An encouraging point is that the best of these fits provide more or less equal quality B( n and simulation results. Certainly, the fit ob­tained here deserves to be used to predict further thermo­physical properties to see if its utility will be maintained. To provide an overall perspective to this work, one should rea­lize that ab initio B( n results were formerly not accurate to better than 30% and this work, as well as unpublished work we have done on the H2 ° dimer, indicates that accuracies of a few percent are now achieveable.

While contributions to the potential surface from the higher-order (octapole, etc.) permanent moments play only a modest role here, this might not be so true for other pair potentials. Consequently, some further knowledge of the ac­curacies of such moments would be worthwhile.

In the experimental work of Cohen and Birnbaum,27 it is seen that the experimental octapole of methane can have uncertainties as great as 20% due principally to the suitabil­ity (or lack of it) of the potential form used.

On the other hand, a theoretical study by Diercksen and Sadlej28 displays Hartree-Fock values which differ by 10% or more. Theoretically, there are two principle sources of error: (i) the basis set used. and (ii) the order of perturbation theory employed. For the former, it is not clear that a suffi­ciently large basis set has been used. In particular, for an operator of rank L (octapole = 3), first-order corrections withsymmetriesashighasL + 1 (Catom) andL (Hatom) have a nonvanishing contlibution. Unfortunately, few if any molecular computer codes exist which can treat symmetries above! For the latter there is some speculation that correc­tions from higher-order perturbation theory may be small,6,28 perhaps several percent. In any case, correlated monomer values are generally not available (these would require separate many-body perturbation theory calcula­tions). Zero-point effects are another reason for the differ­ence between theory and experiment, but these are probably

smaller than the effects mentioned already. For the octapole moment of methane, the basis set of

Szczesniak et al.6 seems to undershoot the best theoretical values28 by about 15%, and these in turn undershoot "ex­periment" by > 6%. Some preliminary work has been done to improve description of the octapole moment by adding a C-atom-centeredf function to the basis set. For thef-expo­nent, Szczesniak et al.6 have used a value of 0.284 which increased the Hartree-Fock octapole moment 2.7%. We have adjusted this exponent to maximize the octapole mo­ment (exponent = 0.500), obtaining a Hartree-Fock mo­ment of2.349 a.u., an 8.6% increase over the value obtained from thef-Iess basis set.6 This is about 5% smaller than a theoretical value28 obtained with a much larger basis set (which, however, has no g C-centered functions). We are now using this new basis for our study of CH4 -H20, where the role of the octapole moment may be larger.

The situation regarding even higher moments (e.g., hexadecapole) is even worse-even more inadequate basis sets are in use, and it is even more difficult to, "unravel" hexadecapole moments from experiment.

ACKNOWLEDGMENT

We gratefully acknowledge partial support of this re­search by the Gas Research Institute, Contract No. 5089-260-1239.

1 D. E. Woon and D. R. Beck, J. Chern. Phys 92,3605 (1990). 'D. E. Woon, P. Zeng, and D. R. Beck, J. Chern. Phys. 93, 7808 (1990). 3D. H. Gay, D. R. Beck, and D. E. Woan, J. Chern. Phys. (submitted). 4K. Szalewicz, S. J. Cole, W. Kolos, and R. J. Bartlett, J. Chern. Phys. 89,

3662 (1988). sUo Niesar, G. Corongiu, E. Clementi, G. R. Kneller, and D. K~ Bhatta­charya, J. Phys. Chern. 94, 7949 (1990).

6M. M. Szczesniak, G. Chajasinski, S. M. Cybulski, and S. Scheiner, J. Chern. Phys. 93, 4243 (1990).

7 G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham, Intermole­cular Forces (Clarendon, Oxford, 1981).

8 S. Wilson, Electron Correlation in Molecules (Clarendon, Oxford, 1984). 9 R. Krishnan, M. J. Frisch, and J. A. Pople, J. Chern. Phys 72, 4244

(1980); R. Krishnan and J. A. Pople, Int. J. Quantum Chern. 14, 91 (1978).

lOW. Kolos, G. Ranghino, E. Clementi, and O. Novaro, Int. J. Quantum Chern. 17,429 (1980).

11 A. J. Sadlej, eoll. Czech. Chern. Commun. 53,1995 (1988). 12S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970). 13 Program, GAMESS, M. W. Schmidt, K. K. Baldridge, J. A. Boatz, J. H.

Hensen, S. Koseki, M. S. Gordon, K. A. Nguyen, T. L. Windus, and S. T. Elbert, QCPE Bull. 10, 52 (1990).

14K. T. Tang andJ. P. Toennies, J. Chern. Phys. 80, 3726 (1984). IS D. E. Woan, Ph.D. thesis, Michigan Technological University, 1988 (un­

published). 16D. R. Beck and D. H. Gay, J. Chern. Phys. 93, 7264 (1990). 17 J. S. Koch, S. Friberg, and T. Larsen, in Zahlenwerte und Funktionen,

edited by H. A. Landolt and R. Bornstein (Springer-Verlag, Berlin, 1962).

13 E.g., W. Meyer, J. Chern. Phys. 58,1017 (1973). 19S. Murad, Program Non Linear, Quantum Chemistry Program Ex­

change, Program No. 357 (1978). 20 A. H. Stroud, Approximate Calculations of Multiple Integrals (Prentice­

Hall, New York, 1972). 21 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory oj

J. Chern. Phys., Vol. 95, No. 12,15 December 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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9114 Gay, Dai, and Beck: Second virial coefficients for methane

Gases and Liquids (Wiley, New York, 1954), p. 420ff. 22S. Murad, D. J. Evans, K. E. Gubbins, W. B. Street, and D. J. Tildesley,

Mol. Phys. 37, 725 (1979). .. 23R. T Pack, J: Chern. Phys. 78, 7217 (1983). 24 J. H. Dymond and E. B. Smith, The Viria/ Coefficients of Pure Gases and

Mixtures (Clarendon, Oxford, 19&0). 2sR. D. Goodwin, Natl. Bur. Stand. (U.S.) Techn. Note No. 654 (1974). 26 A. D. Buckingham and A. J. C. Ladd, Can. J. Phys. 54, 611 (1976). 27 E. R. Cohen and G. Birnbaum, J. Chern. Phys. 66, 2443 {1977). "G. H. F. Diercksenand A. J. Sadlej, Chern. Phys. Lett. 114, 1~7 (1985).

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