Nuclear spin-spin coupling in the methane isotopomers

JOURNAL

OF MAGNETlC

RESONANCE

93,458-47 1 ( 199 1)

Nuclear Spin-Spin Coupling in the Methane Isotopomers JAN GEERTSENAND JENS ODDERSHEDE Department

of Chemistry,

Odense

University,

DK-5230

Odense

M, Denmark

W. T. RAYNES Department

of Chemistry,

The University,

Shefield,

S3 7HF,

United

Kingdom

AND

GUSTAVO E. SCUSERIA Department

of Chemistry,

Rice

University,

Houston,

Texas

77251-1892

Received June 14, 1990; revised December 18, 1990 The coupling constants ‘J(C, H) and ‘J(H, H) of the methane molecule have been calculated as functions of bond-length extension and compression in the vicinity of equilibrium geometry. This has facilitated the prediction of the temperature dependences of these couplings. The calculations were carried out using various polarization propagator methods. There is a very large contribution from electron correlation to both couplings. The bond-length dependence is dominated by the Fermi-contact part of the coupling. ‘J(C, H) is calculated to increase by 0.054 Hz upon increasing the temperature of r3CH4 from 200 to 400 K. This result is less than the observed value of 0.083 Hz due to the neglect of higher-order terms, including those involving the angle dependence of the coupling. 2.1(H, D) in “CH3D is calculated to be virtually temperature independent. The calculated total carbon-proton coupling at 300 K is 126.3 1 Hz, which is only 1 Hz greater than that experimentally observed. The calculated total proton-proton coupling at 300 K is -14.24 Hz, which is numerically greater by about 2 Hz than that calculated from a recent measurement on ‘*CH3D. o 1991 Academic PKS, IX.

Nuclear spin-spin coupling constants are not constant. The small variations in measured values due to solvent effects and intermolecular interactions have been known for many years ( 1). However, even when such effects are absent or have been fully accounted for, there remain variations which, although normally very small in relation to the total coupling, can be attributed to intramolecular phenomena (2). These variations are observed upon changes of temperature and isotopic substitution. For example, the carbon-proton coupling .I( C, H) in 13CH4 gas at low density increases from 125.266 Hz at 200 K to 125.349 Hz at 370 K while J(C, H) in 13CHD3 gas at low density increases from 124.222 Hz at 200 K to 124.3 17 Hz at 370 K (3). Luzikov and Sergeyev (4) found that ‘J( C, H) and *.I( C, H) in acetylene dissolved in acetone solution at room temperature are 248.29 and 49.74 Hz, respectively, while they are 247.78 and 50.03 Hz, respectively, in acetylene-d2 under the same conditions and 0022-236419 I $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

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after multiplication of the measured J( C, D) values by 6.5 144 ( =-yn/~n). Variations of 2J(H, D) in the methane isotopomers CH3D, CH2D2, and CHD3 at different temperatures have recently been reported in a very accurate study by Anet and O’Leary at 500 MHz (5). However, as different solvents were used in that work it is likely that their results, which are generally accurate to better than kO.001 Hz, include variable intermolecular contributions. The intramolecular contributions are due to the sensitivity of the coupling to the molecular geometry. For a given coupling in a molecule there is a spin-spin coupling surface representing the coupling as a function of a full set of independent, internal nuclear coordinates. As most molecules engage only in small excursions from equilibrium geometry it is generally valid to describe that portion of a surface sampled during such excursions as a power series expansion about equilibrium in these coordinates (6). Averaging over the nuclear motion then gives values for the coupling in individual vibration-rotation states ( 7) or, if desired, at any temperature (8). To explain these contributions quantitatively it is necessary to carry out highly accurate calculations of spin-spin coupling constants at several geometries. The fundamental theory and methods of calculation of spin-spin coupling have been reviewed critically by Kowalewski ( 9,lU). It is now generally recognized that in such calculations all four parts of the coupling-Fermi-contact (FC), spin dipolar (SD), orbital paramagnetic (OP), and orbital diamagnetic (OD)-must be computed. Furthermore, the Fermi-contact part is greatly influenced by electron correlation so that good values of J(FC) and it derivatives cannot be obtained at the SCF level. Very recent work using the polarization propagator approach (11) and large basis sets has achieved excellent results for small diatomic molecules. The coupled-cluster singles and doubles polarization propagator approximation (CCSDPPA) gave results for the bond-length dependence of the coupling in the HD molecule which led to almost total agreement between observed and calculated values of J( H, D) at 40 K (12). A second-order polarization propagator approach (SOPPA) also gave greatly improved agreement with experiment for two molecules, CO and N2, in which electron correlation effects on the couplings are substantial (13). For polyatomic molecules the only previously published ab initio work on spinspin coupling surfaces is for J( C, H) and J( H, H) in methane ( 14). This calculation showed that there are very marked geometry dependences for J( FC), J( SD), and J( OP) for both couplings. However, these results were obtained at the SCF level and so electron correlation was neglected in calculating J( FC), which forms the dominant part of each coupling. In the present paper we report the results of RPA (random phase approximation ), SOPPA, and CCSDPPA calculations for the couplings J( C, H) and J( H, H) in CH4. All four parts are calculated as functions of the symmetric stretching and contraction of the four C-H bonds. This is sufficient to account for the average geometry dependence for CH4 and CD4 to first order in internal symmetry coordinates. The dependence on asymmetric stretching and angle bending enters the vibrational average only at second and higher orders for these isotopomers. METHOD

Three of the terms contributing to the nuclear spin-spin coupling constant (FC, SD, and OP) are of second order in the magnetic perturbation and can therefore be expressed as a sum over states,

460

GEERTSEN

ET

AL.

where V, and VNr refer to one of the three perturbations nuclei N and N’. JNNr can be expressed as

FC, SD, and OP for the

J NN’ = f (( viv; ~w>:)E=o,

[21

where

((G

Kv’))E = C’

is the polarization pressed as

(01w+(4

n

KV’IO)_ (01Kdn)(nl VNIO)

E - E,, + E.

E i- En - E.

I

[31

propagator or linear response function. This function can be ex(( VN; I/N+)

=./I v.iv, EV’(E)-‘.f(

VW; ~9,

141

where P(E)-’ is the so-called principal propagator. It depends on the electronic structure of the molecule and on the level of approximation at which the calculation (RPA, etc.) is done. However, it is independent of V, and l’,,r and only the E = 0 limit of P(E) is needed for the calculation of JNNt (see Eq. [ 21). Furthermore, once P( E = 0) is known it is straightforward to obtain all three second-order contributions. This gives a clear advantage over a finite-field calculation, where a full calculation is needed for every contribution to the coupling. The equation of motion of the propagator is solved perturbatively with the fluctuation potential-that is, the electronic repulsion-minus the Fock potential, as the perturbation. The zeroth-order approximation is defined as the Hat-tree-Fock (HF) approximation where both ground IO) and excited states In) in Eq. [ 31 are singledeterminantal wavefunctions. The consistent first-order approximation RPA adds correlation in both ground and excited states and further correlation is added in SOPPA. The finite-field SCF method gives the same result as does RPA while there is no finitefield equivalent to SOPPA. Thus in the finite-field framework one would define RPA as the SCF solution, i.e., one without correlation. However, within the polarization propagator formulation RPA is first order in the fluctuation potential. The difference between the two ways of viewing RPA is connected with how one expresses the response of the system (in the present case, the change in magnetic moment of one of the nuclei) due to the “external” perturbation (i.e., the other nuclear spin). In RPA one expresses JNNTin terms of the unperturbed states while in the finite-field SCF method the coupling is given in terms of the fully relaxed orbitals. One is thus referring to order of perturbation theory with respect to different zerothorder Hamiltonians. We have also calculated coupling constants using the coupled-cluster improved version of SOPPA referred to above, viz. CCSDPPA (1.5). Like SOPPA, this is a second-order method but, by using the coupled-cluster wavefunction as the reference state rather than a Rayleigh-Schriidinger expansion, several extra series of diagrams are included to infinite order. This kind of improvement is important in cases where normal Rayleigh-Schrodinger perturbation theory is only slowly convergent. However, even for well-behaved molecules like HD, CCSDPP.A gives a significant improvement over SOPPA for coupling constants when high accuracy is the aim ( 12).

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Recent reviews (12, 16) describe the polarization propagator method in detail. Explicit expressions needed for the polarization propagator calculations of coupling constants have also been given ( 17). All coupling-constant calculations in this work have been performed with a program written by one of us ( 28). The OD contribution is a first-order property and therefore cannot be expressed by Eq. [ 11. We calculate it simply as an SCF average value using the method of Matsuoka and Aoyama ( 19) since previous experience indicates that J( OD) depends only slightly on electron correlation (20). THE BASIS SET

The choice of a one-electron basis set is most important for accurate spin-spin coupling constants. Accordingly, to determine an optimal basis set, we have performed a series of calculations using extensions of the van Duijneveldt basis set (21). All calculations were performed at the geometry used by Lazzeretti and Zanasi (22). The basis sets tested are given in Table 1 and the corresponding coupling constants appear in Table 2. From the first five rows of Tables 1 and 2 it can be seen that contraction of the s basis functions on C and H is not possible while a modest contraction of the p set on C is permissible. As first noted by Schulman and Kaufman (23) and as shown for HD ( 12)) it is important to include large exponent s functions in the basis set in order to obtain a reliable J( FC) . From the B7-B8 and B 14-B 15 calculations it is seen that the largest exponent needed for sc is 40 X lo6 while 80,000 is the limit for su . It is to be noted that the addition of very tight functions has no effect on the total energy, only on the densities at the nuclear positions. We also added diffuse s functions (B 11) and pc functions as well as tight pc functions to the basis set with no effect on the computed values of the coupling constants. The results of calculations B 16-B 19 show that d functions on the carbon are needed and that exponents less than 0.04 and greater than 40 can be omitted. A closer study of the results of Table 2 reveals that the addition of dc functions has an effect on the noncontact terms relatively larger than that of the other functions so far discussed. With the hope of adding polarization functions on the protons we have investigated the possibility of contracting the s functions with the largest exponents, bearing in mind the results of B3 to B5. Accepting a basis set error of about 0.2 Hz, one sees that the contractions of B20 and B22 can be used. Finally, the calculations using the basis sets B24 and B25 show that two p-polarization functions on H are needed. We have also performed one calculation with basis set B24 plus a single d function on H. This had little effect on the computed coupling constants. In summary, we have shown that the final [ lOs5p4d/ 5.~2~1 basis set consisting of 97 CGTOs is converged with respect to the calculation of both J( C, H) and J( H, H ) . The remaining basis set error is estimated to be of the order of +0.2 Hz. RESULTS

Values of J(C, H) at the different levels to those of Lazzeretti quality. As expected,

and J(H, H) for methane at equilibrium geometry calculated of theory are given in Table 3. The RPA results are very similar et al. (14), demonstrating that both basis sets are of the same electron correlation reduces the numerical values of both con-

462

GEERTSEN

ET

TABLE Descriution Label Bl B2 B3 84 B5 B6 87 B8 B9 BlO Bll B12 B13 B14 B15 B16 B17 818 B19 B20 B21 B22 B23 B24 B25

Composition [6s5pld/3slp]6 [6s7pld/3slp]’ [1 ls5pld/3slp]* [6s5pld/5slplb [I ls5pld/5slp]b B5 + (E = 125,000, & = 225) B6 + (E = 900,000) B7 + (PC = 6 X 106) B8 + (Pc = 40 x 106) B9 + (pc = 300 x 106) B9 + (s’sc = 0.044) B9 + (s’s” = 1,600) B12 + (pu = 11,200) B13 + (pi = 80,000) B14 + (rn = 500,000) B14 + (SC = 0.15, 2.4) B16 + ({$ = 10) 816 + (<$ = 0.04) B17 + ({$ = 40) B17 with 15sc + 1Osc’ B17 with 15sc + 9sc’ B20 with 9s, + rnc B20 with 9s” + 5s,’ B22 + (39” = 0.35) B22 - (3”, = 1.5) + (r”, = 0.25, 0.75, 2.25)

1

of One-Electron CGTOs

AL.

Basis Sets -hWa

WCY

51 57 56 59 64 69

-40.2124 -40.2124 -40.2 126 -40.2125 -40.2127 -40.2132

122.38 122.38 122.24 122.38 122.24 124.42

0.4267 0.4268 0.4267 0.4343 0.4606 0.4606

70 71 72 73 73 76 80 84 88 96 102 102 108 97 96 85 81 97 109

-40.2132 -40.2132 -40.2132 -40.2132 -40.2132 -40.2133 -40.2133 -40.2133 -40.2133 -40.2 147 -40.2 147 -40.2 147 -40.2 147 -40.2 147 -40.2 147 -40.2147 -40.2 147 -40.2151 -40.2 150

125.18 125.46 125.57 125.61 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57 125.57

0.4606 0.4606 0.4606 0.4606 0.4606 0.4735 0.4772 0.479 1 0.4795 0.4795 0.4795 0.4795 0.4795 0.4795 0.4795 0.4798 0.4796 0.4797 0.4797

’ In atomic units (au.) with rCH = 2.06 1731 au., &c) and @ru) are SCF densities at the nuclear positions (1 au. = 0.529177249 X lo-” m). b Contraction of the (11 ~7~15s) basis set of Ref. (21) with added polarization functions with exponents <$ = 0.6 and & 1.5. c The s basis functions with the largest exponents are contracted to a singles function using as contraction coefficients the SCF eigenvectors from a calculation in the uncontracted basis set.

stants. Recent experimental work (3) has shown that the equilibrium value ‘J, of J( C, H) is 120.78 (kO.05 ) Hz. Therefore, assuming that our RPA value is close to the finite-field Hartree-Fock limit for J( C, H), the CCSDPPA result has recovered about 91% and our SOPPA result about 79% of the correlation contribution. For the 2J( Hj H) coupling the corresponding changes are 88 and 79%, respectively. As found for HD (12), the additional correlation yielded by the CCSDPPA method is small but important as it always improves the calculated couplings. Other features of the results in Table 3 are (i) the very large (percentage) correlation contribution to the SD term of J( C, H), (ii) the nearly constant values of the (singlet) OP term for both

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ISOTOPOMERS

TABLE 2 Nuclear Spin-Spin Coupling Constants” of CH4 (in Hertz) as a Function of the One-Electron Basis Sets in Table 1

‘W, W

2J(H, W

Basis set

FC

Total

FC

Total

Bl B2 B3 B4 B5 B6 Bl B8 B9 BlO Bll B12 B13 B14 B15 B16 B17 Bl8 B19 B20 B21 B22 B23 B24 B25

119.81 119.76 115.06 119.12 113.25 122.21 122.95 123.23 123.34 123.38 123.33 126.81 127.79 128.31 128.43 127.99 128.17 127.98 128.16 128.18 128.27 128.28 128.03

121.54 121.44 116.79 120.84 114.97 123.93 124.68 124.95 125.06 125.10 125.06 128.53 129.51 130.03 130.15 129.67 129.83 129.67 129.85 129.83 129.92 129.93 129.68 129.41 129.30

-15.36 -15.36 -15.35 -13.97 -13.97 -15.59 -15.59 -15.59 -15.59 -15.59 -15.58 -16.52 -16.76 -16.90 -16.93 -16.47 -16.47 -16.46 -16.47 - 16.47 -16.47 -16.40 -16.68 -16.02 -15.93

-15.23 -15.17 -15.23 -13.84 -13.83 -15.45 -15.45 -15.45 -15.45 -15.45 -15.44 -16.38 - 16.62 -16.76 -16.79 -16.12 -16.16 -16.11 -16.11 -16.11 -16.11 -16.04 -16.32 -15.61 -15.52

127.74

127.58

’ SOPPA results with rc- = 2.06 173 1 a.u.

couplings, and (iii) the characteristic mutual cancellation of OP and OD terms for geminal proton-proton couplings (25). We now consider variations of the four bond lengths with the interbond angles retaining their equilibrium value of 109”28’. We define the totally symmetric stretching coordinate S, by S, = f(rl

+ r2 + r3 + r4),

[51

where rj denotes an extension of the ith C-H bond. By always having rl = r2 = r3 = r4 = r we render all other stretching coordinates (26) zero and, of course, S, = 2r. Results for ‘J(C, H) and 2J(H, H) as r is varied from -0.2 to +0.2 A ( 1 A = lo-” m) are given in Figs. l-4 for each contribution to the coupling. Figure 1 shows the dependence of the Fermi-contact contribution to J( C, H) on S, at the three levels of theory. For S, = 0 the results are given in Table 3. However, as the figure makes clear, the RPA derivative with respect to Sr is also much too high at each of the chosen values of S, and this leads to J(C, H) being overestimated by a factor of about two

464

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TABLE 3 Effect of Electron Correlation on the Nuclear Spin-Spin Coupling Constants of CH, (in Hertz)’

‘JC, W Method

FC

RPA SOPPA CCSDPPA

155.57 126.62 122.12

Expt.

‘JW, W

SD

OP

Total *

-0.23 -0.02

1.39 1.48 1.50

156.98 128.32 123.87

-0.01

FC

SD

OP

Total ’

-27.63 -16.04 -14.70

0.44 0.34 0.33

3.63 3.62 3.60

-27.10 -15.67 -14.31

120.78(k0.05)d

- 12.564(-C0.004)e

a At the experimental (24) equilibrium geometry, rcn = 2.05 1864 au. (1.0858 A). * Including the OD term of 0.25 Hz. ‘Including the OD term of -3.54 Hz. d Value for r, = 1.0858 A. See Ref. (3). ’ Value calculated from ‘J(H, D) for CH,D in CCL, solution at 22°C. This may include both intramolecular and intermolecular effects. See Ref. (5).

for S, = 0.4 A. The SOPPA and CCSDPPA results are much closer but there is a clear superiority ofthe CCSDPPA result over the SOPPA result which becomes more marked as S, increases. Figure 2 shows the dependence of the Fermi-contact contribution to J( H, H) on S, at the three levels of theory. Again the (numerical) value is grossly overestimated by the RPA calculation while the SOPPA and CCSDPPA results are much closer to each other but still different with an increasing difference as S, increases.

140 120 100

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Sl(angstroms) FIG. 1. Dependence of the Fermi-contact contribution to J( C, H) on the symmetric stretching coordinate S, . Results are given at the RPA, SOPPA, and CCSDPPA levels of theory.

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CCSDPPA

L-2

-0.4

#I(_

-0.2

0

0.2

I 4

S,(angstroms) FIG. 2. Dependence of the Fermi-contact contribution to J( H, H) on the symmetric stretching coordinate S, Results are given at the RPA, SOPPA, and CCSDPPA levels of theory.

In this case the RPA derivative with respect to S, is not only overestimated but also has the wrong sign. Figure 3 shows the spin-dipolar contributions for both couplings plotted against S, . Yet again RPA-calculated contributions to both J( C, H)-see lowest curve in Fig. 3-and J(H, H)-see highest curve in Fig. 3-are numerically overestimated at almost all values of S,, especially for large S, . Figure 4 shows the two orbital contributions for both couplings plotted against Si . The lower part of the figure shows the SCF calculation of the orbital diamagnetic contribution for J( C, H) plotted against S, and the RPA and CCSDPPA calculations of the orbital paramagnetic contribution to J(C, H) plotted against S, . The SOPPA curve is not plotted as it is almost coincident with (but slightly lower than) the CCSDPPA curve. In the upper part of Fig. 4 plots of the SCF orbital diamagnetic and CCSDPPA orbital paramagnetic contributions to J( H, H) plotted against Si are shown. It is to be noted that it is the numerical value of (the negative) 2J(OD) which is plotted. These two plots make clear that it is not only at equilibrium geometry (see Table 3 ) but also for a wide range of Si values that the two orbital contributions almost exactly cancel one another. The physical reason for the cancellation of these contributions to the geminal coupling is not understood. The RPA and SOPPA results for the orbital paramagnetic contribution to J( H, H) are not plotted in Fig. 4 as they are almost identical to the CCSDPPA results, showing that electron correlation is not important for the orbital paramagnetic term. In making comparisons between Figs. 1 through 4 it is important to note the changes of vertical scale from figure to figure. For the highest level of theory, results are presented explicitly in Table 4. It is clear from the figures that it is the variation of the FC terms which determines the variation of the total couplings. It is also clear that vibrational corrections in

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‘J(H,H) *J(H,H)

\ RPA\ ‘JGH)

-0.4

-0.6

‘J(C,H)

I/.-0.4

-0.2

0

Y 0.2

0.4

S,(angstroms) FIG. 3. Dependence of the spin-dipolar contribution to J(C, H) and J( H, H) on the symmetric stretching coordinate S, . Results are given at the RPA, SOPPA, and CCSDPPA levels of theory.

SOPPA and CCSDPPA will be considerably smaller than those in RPA. Perhaps the most striking effect of correlation beyond RPA is the sign change of the derivatives of 2J( FC), which (see below) gives a positive vibrational contribution to 25( H, H) at first order instead of a negative one as previously obtained at the RPA level (26). It is also to be noted that all contributions to the coupling constants are monotonic functions of S, and that, except for *J( FC), the derivatives have the same sign at the three levels of theory. VIBRATIONAL

AVERAGES

For a small but arbitrary displacement of the four protons of methane from equilibrium the carbon-proton coupling constant involving proton 1 can be written (6b) J(C, H1) = Je(C, Hl) + JI(C, Hl IS, +

J3z(C

I-II

MS3x

+ JdC,

+

S3,

+ 83,)

Hl)(S,,

+

&,

+

s4,)

161

to first order in the symmetry coordinates. Similarly the coupling constant involving protons 1 and 2 can be written (6~) J(H1, H2) = J,(Hl,

H2) + J,(Hl,

H2)S1 + J,,(IIl, + J,AHl,

H2)S3z

H2)$, +

J4AH1,

H2P4,,

171

also to first order in the symmetry coordinates. IJpon averaging over the nuclear motion only the totally symmetric coordinate S1 does not average to zero for CH4, CD4, and CT4 so that only the first two coefficients in each of Eqs. [ 61 and [ 71 are

COUPLING

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‘J(OP) ‘J(OD) 4

S,(angstroms) FIG. 4. Dependence of the orbital diamagnetic and orbital paramagnetic contribution to /(C, H) and J( H, H) on the symmetric coordinate S, Orbital diamagnetic results were calculated at the SCF level. The orbital paramagnetic contribution to .I( C, H) is plotted at both the RPA and CCSDPPA levels. The orbital paramagnetic contribution to J( H, H ) is plotted for the CCSDPPA level only (see text). Note that it is the numerical value of the negative orbital diamagnetic contribution to J( H, H) which is plotted against S, ; this has been done to emphasize the near mutual cancellation of the orbital terms over a wide range of S, .

needed to first order for these isotopomers. Fitting the CCSDPPA results in Table 4 for the total couplings to an expression of the form .I, + J,S, + 1 .I,, S: gives (in hertz ) J(C, Hl) = 123.56 + 88.983, + 14.94s: t81 and

[91

J(H1, H2) = -14.30 + 2.5OSr + 3.323:. To proceed further we convert to reduced normal coordinates qr using the relation s, = c z:‘;qr

r

[lOI

defined by Hoy et al. (27). The relation between S, and qr is strictly nonlinear but Eq. [lo] is valid for terms linear in the qr although the coefficients J, (C, Hl ) and J, (H 1, H2) do make contributions at second order. For CH4 and CD4 there are four distinct modes, only one of which, ql, does not average to zero. Thus we need only &fand(q,)T.Thelatterisgivenby(8)

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TABLE 4 Nuclear Spin-Spin Coupling Constantsa of C& (in Hertz) as a Function of the Symmetric Stretching Coordinate S,

‘JCC,W s, -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

r(A)

-0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20

‘JW, W

-

FC

SD

OP

OD’

Total

FC

SD

OP

OD’

Total

80.07 92.17 102.76 112.57 122.12 131.81 141.92 152.67 164.25

0.48 0.32 0.19 0.09 -0.0 1 -0.08 -0.14 -0.19 -0.24

2.37 2.12 1.88 1.68 1.50 1.34 1.21 1.09 0.99

0.91 0.66 0.48 0.34 0.25 0.18 0.13 0.09 0.06

83.83 95.26 105.31 114.68 123.87 132.25 143.11 153.65 165.06

-15.92 -15.62 -15.32 - 15.02 - 14.70 -14.34 -13.91 -13.35 -12.60

0.56 0.49 0.43 0.37 0.33 0.30 0.28 0.26 0.24

7.08 5.91 4.98 4.22 3.60 3.09 2.67 2.31 2.01

-6.57 -5.58 -4.77 -4.10 -3.54 -3.07 -2.67 -2.33 -2.04

-14.85 -14.80 -14.69 -14.53 -14.31 -14.02 -13.64 -13.11 -12.39

a CCSDPPA results. b r, = 2.05 1864 au. (1.0858 A). c SCF values (see text).

where T is the absolute temperature, w, is the frequency (in cm-‘) of the totally symmetric mode, and 02, w3, and o4 are the frequencies of the other three modes labeled s with degeneracies g, equal to 2, 3, and 3, respectively. The quantities k, 1r and k,, are cubic anharmonic force constants in the Nielsen formulation of the potential energy function (28), aI is the first derivative of the moment of inertia with respect to ql, and Ice) is the equilibrium value of the moment of inertia. In carrying out the numerical calculations, we have used previously tabulated data (29) taken from the experimental results of Gray and Robiette (24). For & 1, given by z;

= $(L; + L; + L; - LA),

[I21

we obtain -0.105 155 A for ‘*CH, and 13CI& and -0.088442 A for ‘*CD4 and 13CD4. Results for the first-order vibration-rotation contribution to the one-bond coupling constants of 13CH4 and 13CD4 at various temperatures are given in Table 5. For 13CH4, results are given at the RPA, SOPPA, and CCSDPPA levels. The calculated increases in ‘J(C, H) for 13CH4 gas over the range 200-400 K are 0.114 Hz (RPA), 0.063 Hz (SOPPA), and 0.054 Hz (CCSDPPA). The observed increase over the range 200370 K (3) is 0.083 Hz, which is closer to the SOPPA result than to the others. However, the CCSDPPA result is the best result to first order with the additional contributions coming from second-order terms. The only second-order contribution calculable at the present is that due to the quadratic term 14.94s: in Eq. [ 81. This gives a large contribution of 0.786 Hz, which is the same at all temperatures and hence makes no contribution to the temperature dependence. We believe that it is the low-frequency modes v2 and u4 which make substantial contributions at second order as they are known to do for other properties (29, 30) and which contribute the additional part of the temperature dependence. For gaseous 13CD4, Table 5 gives the results at CCSDPPA level assuming that Yn = yn for purposes of comparison with the 13CH4

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TABLE 5 Calculated First-Order Contributions, in Hertz, to the One-Bond Coupling in 13CH4and “CD., Over the Temperature Range 1SO-500 K AJ(C, H 1) (13CH4)

AJ(C, Hl) (‘%D,)

T 6)

RPA

SOPPA

CCSDPPA

CCSDPPA

150 200 300 400 500

5.051 5.083 5.146 5.197 5.234

2.810 2.829 2.863 2.892 2.912

2.398 2.414 2.443 2.468 2.485

1.753 (0.269) 1.768 (0.27 1) 1.773 (0.275) 1.810 (0.278) 1.82 1 (0.280)

Note. For r3CH.,, results are given at the RPA, SOPPA, and CCSDPPA levels. For “CD4, results are for the CCSDPPA level only. 13CD., results not in parentheses assume that the magnetogyric ratio for the deuteron is the same as that for the proton while “CD4 results in parentheses have been obtained by dividing by 6.5 144 to give results appropriate to the experiment.

results and also, in parentheses, as actual contributions. The calculated first-order increase in ‘J( C, D) over the range 200-400 K is 0.007 Hz, which is smaller than but probably not experimentally distinguishable from the observed (3) increase of 0.0 12 Hz over the range 200-370 K. Again we attribute the difference to second-order terms. Table 6 presents similar results for the two-bond couplings in 12CH4 and 12CD4. The most striking result here is that the RPA calculation gives a negative sign for the first-order contributions and predicts that the coupling constant diminishes by 0.009 Hz upon raising the temperature from 200 to 400 IL The CCSDPPA calculation gives a small positive contribution which increases by 0.001 Hz upon increasing the temperature over the same range. Of course, the only observable form of this coupling is as 2J( H, D) in the mixed isotopomers CHj D, CH2 D2, and CHD3. The results will be very similar to those of 12CHq, however, after one allows for the magnetogyric ratio

TABLE 6 Calculated First-Order Contributions, in Hertz, to the Geminal Coupling in r2CD, Over the Temperature Range 150-500 K AJ(H1, H2) (“CH,) TW

RPA

SOPPA

150 200 300 400 500

-0.40 1 -0.404 -0.409 -0.413 -0.416

+0.014 +0.014 +0.015 +0.015 +0.015

AJ(H1, H2) (“CD,) CCSDPPA +0.067 +0.068 +0.069 +0.069 +0.070

(0.010) (0.010) (0.011) (0.011) (0.011)

CCSDPPA 0.049 (0.008) 0.050 (0.008) 0.05 1 (0.008) 0.05 1 (0.008) 0.05 1 (0.008)

Note. For ‘rCH4, results are given at the RPA, SOPPA, and CCSDPPA levels. For 12CD4, results are for the CCSDPPA level only. Results in parentheses have been obtained by dividing by 6.5 144 and therefore are closely approximate to those for “CH3D for ‘*CH, and “CHD3 for “CD4.

470

GEERTSEN

ET AL.

as shown in parentheses with the CCSDPPA results in Table 6. We calculate a very small increase of less than 0.00 1 Hz upon raising the temperature from 200 to 400 K. Thus we predict a slight decrease in the numerical value of 2J( H, D) . This calculated result disagrees in sign with the observations of Anet and O’Leary (5) that 2J( H, D) in CH3D, CH2D2, and CHD3 increases in magnitude by about 0.005 Hz upon increasing the temperature from -50°C to 22°C. However, their measurements were made in acetone and it is very probable that intermolecular effects which are known to increase the magnitudes of coupling constants ( I ) are predominant. Again, secondorder contributions of the low-frequency modes u2 and v4 may make an extra contribution. The term 3.32 ST of Eq. [ 91 contributes 0.175 Hz to the coupling-even larger than the first-order CCSDPPA contribution. However, this is the same at all temperatures and so plays no part in the temperature dependence. The results in Table 6 for 12CD4 without parentheses assume that yn = TH for both nuclei for easy comparison with the 13CH4 results. Those inside parentheses allow for one nucleus with the proton magnetic moment to give approximate predictions for 12CHD Adding ihe vibrational corrections in Tables 5 and 6 at the CCSDPPA level to the CCSDPPA equilibrium couplings in Table 3 gives ‘J(C, H) = 126.31 Hz and 2J( H, H) = - 14.24 Hz for r3CH4 and 12CH4, respectively, at 300 K. The ‘J( C, H) value is only slightly larger than the 125.306 Hz obtained experimentally at this temperature (3). Thus due to the cancellation of errors, J, being slightly too large and A J( C, H 1) being slightly too small, we find better agreement with experiment than we would anticipate from the results in Table 3. Anet and O’Leary (5) give 2J( H, H) = - 12.564 Hz for 12CH4 in CC14. However, this result is obtained from the mixed isotopomers in Ccl4 solution and so neglects any nuclear motion contributions. CONCLUSIONS

Using the polarization propagator method we have calculated the spin-spin coupling constants of methane. RPA (equivalent to finite-field SCF) gives unacceptable couplings while both SOPPA and CCSDPPA give much improved values. There is a small but significant improvement by CCSDPPA over SOPPA. The necessity of using saturated s ,p, d basis sets is clear-see Tables 1 and 2. The nuclear motion corrections and temperature dependences are computed for both couplings by considering the first-order dependence of ‘J and 2J with respect to the totally symmetric stretching coordinate S,, which is the only displacement coordinate to contribute upon averaging to this order. As expected, the temperature dependence of the couplings is dominated by the geometrical dependence of the Fermicontact term. The nuclear motion corrections are considerably smaller in the more correlated calculations than in the RPA and this is particularly striking for the ‘J coupling, where the RPA method gives the wrong sign for the derivative with respect to S, and hence the wrong sign for the temperature dependence. The change of ’ J( C, H) is calculated at the CCSDPPA level to be 0.054 Hz over the 200-400 K range, which is most but not all of the observed (3) change. The remaining part is attributable to second-order terms. By contrast the change in 2J( H, D) is very small and undetectable by present experimental technique over the same range due to the very small geometry dependence of this coupling.

COUPLING

IN

METHANE

ISOTOPOMERS

471

The total computed couplings, including the first-order rovibrational corrections, are found to be very close to the experimental values. The results of this paper represent easily the most accurate theoretical couplings so far for the methane molecule. ACKNOWLEDGMENTS This project was supported by grants from the Danish Natural Science Research Council (Grant 1 l-6844) and the University of Sheffield Research Fund. The authors thank Mr. M. Grayson of Sheffield University for computational assistance in the fitting procedure. Support from the Welsh Foundation to one of us (G.E.S.) is gratefully acknowledged. REFERENCES

1. M. BARFIELD 2. 3. 4. 5. 6. 7. 8. 9. 10. II.

12. 13.

14. 15.

16. 17.

18. 19. 20.

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

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