The calculation of molecular spin-rotation constants

Chemical Physics 105 (1986) l-6 North-Holland, Amsterdam

THE CALCULATION Ronald

D. BROWN

Department

of Chemistv,

Received 13 November

OF MOLECULAR and Martin

SPIN-ROTATION

CONSTANTS

’

HEAD-GORDON

Monarh University, Clayton, Victoria 3168, Awtralia 1985

It is feasible to compute the spin-rotation tensor U for small molecules with useful accuracy (to within 10% for major components of U) by using the coupled Hartree-Fock method and moderately large basis sets (double-zeta plus polarization plus additional diffuse functions). Alternative simpler procedures (e.g., use of the geometric approximation or the Sadlej approximation) are less satisfactory. Calculations on eight small molecules containing first-row elements and/or phosphorus are presented in support of the above conclusions.

of calculation

1. Introduction

2. Methods

Spin-rotation interaction is a property of molecules that include one or more atomic nuclei with non-zero spin. It results from a term in the molecular hamiltonian representing the interaction of a nuclear magnetic moment with the small magnetic field generated by molecular rotation that arises because the electrons do not follow exactly the rotation of the molecular frame. The contribution of this effect to the molecular hamiltonian is

The earliest attempts to compute second-order properties such as spin-rotation interaction constants involved the derivation of expressions for the energy perturbation by second-order perturbation theory. Early calculations by Wick [l] gave poor results and in general one cannot expect to obtain satisfactory predictions because the perturbation expression is an infinite sum with poor convergence properties, the sum being unpredictably influenced by continuum states etc. Flygare [2] developed the formulation so that the spin-rotation constants are expressed in terms of an average molecular excitation energy, A E, the average value of l/r3 for the molecular electron distribution at the spinning nucleus, and elements of the density matrix. Numerical results for several small molecules for U,, were in error by up to an order of magnitude. The coupled Hartree-Fock (CHF) method, used by Stevens et al. [13] for calculations on LiH, overcomes these difficulties. Our investigations, reported below, show that applied to spin-rotation interaction calculations, it yields results of accuracy sufficient to be helpful in assignment and analysis of hyperfine multiplets in microwave spectroscopy [4,5], but for a molecule with 2n electrons and wavefunctions described by m basis functions there are n(m - n) coupled linear equa-

HSR = -I;U.J,

(I)

where 1, is the nuclear spin of the a th nucleus and U is the 3 X 3 tensor of spin-rotation coupling constants. Splittings of spectral lines in the microwave region produced by this interaction are sufficiently small that they have been observed for only a few molecules. Perhaps for this reason not many attempts have been made to predict this molecular property by ab initio molecular orbital calculations, although greater attention has been paid to the calculation of NMR chemical shifts, a closely related property. The present study is aimed ‘at developing a reliable (lo-20% accuracy) method of predicting this molecular property by ab initio MO calculations on polyatomic molecules. i Present address: Chemistry Department, Mellon Institute (CMU), 440 5th Avenue, Pittsburg, PA 15213, USA.

0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

2

R. D. Brown. M. Head-Gordon / Molecular spin - rotation constants

constants. It is possible to compute the electronic component of U, i.e. U,, from the paramagnetic component of the magnetic shielding tensor, at,, if the gauge origin for u is located on the nucleus of interest:

tions to be solved so that the method becomes computationally unmanageable except for small molecules (or inadequate basis sets). To avoid the computational difficulties associated with the CHF equations, the uncoupled Hartree-Fock (UCHF) method, in which the coupling terms are neglected, has been used [6] but its performance is unsatisfactory. It has been proposed that the UCHF results can be improved by adopting the so-called geometric approximation (GA) [6-lo]. In this procedure each molecular orbital composing the first-order perturbed wavefunction in the UCHF approximation is multiplied by a scaling factor selected to minim&e the second-order energy (i.e. the spin-rotation interaction energy), it having been shown [7] that this is an improved approximation to the CHF perturbed orbitals. Sadlej [ll] has proposed that for the calculation of mixed second-order properties such as spin-rotation interaction energies two scaling factors, one corresponding to each perturbation, should be determined. Here we investigate the reliability of both the (GA) and the Sadlej geometric approximation (SGA) for some small molecules. Although only few calculations of spin-rotation constants have been reported previously (table l), considerably more attention has been paid to the calculation of nuclear magnetic shielding

u, = (2g,p,mco,/lO%e)

l

I-‘,

(2)

where U, is in Hz, up in ppm, pN is the Bohr magneton, g, value of nucleus a and I -’ in the inverse inertia tensor of the molecule. Some conclusions from our present study are therefore applicable to the problem of ab initio calculation of gas-phase NMR chemical shifts.

3. Resume of previous calculations of spin-rotation constants Table 1 lists results reported hitherto in the literature. The values computed by the sum-overstates technique of Flygare [2], which in addition to the use of the “average excitation energy” approximation, neglects multicentre integrals, are unreliable both with respect to the absolute magnitudes of values, which can be in error by up to a factor of 7, and with respect to relative magnitudes of different diagonal components of U. The only reported calculations of spin-rotation

Table 1 Previous SRC calculations Molecule

Nucleus

Component

M(calc) (kHz)

M(obs) (kHz)

HF

19F

perp

-2137

-305

LiF

19F

perp

-116

F,

19F

perp

-177

-153

CH,O

13C

- 217

CH,O

CH,NH

LiH

-37.3

[21

(7)

PI 121 1121

:5:

-29

-127.0 -19.8

(3) (3)

cc

-25

-7.7

(3)

“0

z;

398 50

371 29

(10) (10)

cc

47

2

(10)

14N

;;

-

-87 15

49.5 (47) - 9.7 (27)

cc

-13

- 1.2 (24)

‘Li

9.4

H

- 9.14

10

-8

Ref.

(2)

(1)

(1)

WI

WI

131

R.D. Brown, M. Head-Gordon

constants based on the CHF method are for LiH [3]. The values were within 10% of the experimental values obtained by molecular beam spectroscopy. If we are to judge from more extensive calculations of nuclear magnetic shielding [13-201 then the CHF method yields values within lo-20% of experiment but very large basis sets had to be used, even for very small molecules. Smaller basis sets led to highly gauge-dependent results. Various expedients for improving this situation for more modest basis sets have been proposed, including the use of gauge-invariant atomic orbitals (for example, ref. [20]), or the use of the geometric approximation. For the latter the few reported calculations [17,18] appear to be very satisfactory, leading to the suggestion that for larger molecules the UCHF/GA method will be more fruitful than attempting to employ the CHF scheme. The SGA method does not appear to have been used hitherto in specific calculations.

4. Details of calculations To test the practicability of computing useful values of spin-rotation constants for polyatomic molecules of moderate size we have developed the program COLUMBO that takes Hartree-Fock LCAO molecular orbitals and two-electron integrals over MOs from a modified version of the GAUSSIAN 80-UCSF program [21] and computes spin-rotation constants using the CHF method and the two GAS. All integrals are computed analytically using the properties of gaussian basis functions. Integrals involving angular momentum operators are reduced to linear combinations of integrals over dipole moment and electric field operators, drawing upon routines extracted from the properties package of the POLYATOM program. The CHF equations are solved as a set of coupled linear simultaneous equations to yield the perturbed MOs. Finally the resultant U tensor is transformed to the molecular principal inertial axis system for comparison with spectroscopically determined values. The program also yields nuclear magnetic shielding tensors.

/ Molecular spin -rotation

5.

3

constants

Results and discussion

As a preliminary study of the size and composition of basis set needed to achieve results of useful accuracy and of the way in which the choice of gauge origin can influence the size of the error in the use of a finite basis set we computed the perpendicular component of the nuclear magnetic shielding tensor for HF, using the experimental bond length of 1.7329 au and the following basis sets: (1) STO-4G minimal sp basis set [22]. (2) 4-31G split valence sp basis [23]. (3) [5s3p ]3s] double-zeta (DZ) basis [24]. (4) [5s3pld ]4slp] DZ + polarization basis [24, 251. (5) [5s3pld ]3slp] + Hs(O.07) lp(O.05) (ls(O.06) lp(O.25)] corresponding to set (4) with additional diffuse functions [13]. (6) [5s3p2d I3slp] + [2s2p 12~2~1, the additional diffuse functions being those suggested by Holler’ and Lischka [13]. Figs. 1 and 2 show the variation of diamagnetic shielding at F and H calculated with the above

1 2 3 4 BASIS SETS IN ORDER OF INCREASlffi

Fig. 1. Variation for F in HF.

of diamagnetic

5 6 QUALITY

shielding with basis set used

R. D. Brown, M. Head-Gordon / Molecular spin -rotation constants

4

J

2 3 4 5 SETS IN ORDER OF INCREASING SIZE

Fig. 2. Variation of diamagnetic Shielding at H, gauge on F.

6

shielding with basis set used.

basis sets. The values for u,(F) rapidly converge, showing that even small basis sets yield good results, which is to be expected since a, is a first-order property with a (l/r) dependence. The results for u,(H) do not converge as well; basis (4) is needed for good results. For the paramagnetic shielding, up(F) and u,(H) (figs. 3 and 4), one needs to go to basis (5) to get within 10% of the large-basis-set value. It is also .apparent that the GA and SGA methods provide reasonably good values for sets (4), (5) and (6), F being a more challenging nucleus than H. Overall we conclude that a basis set at least of quality (5), i.e. including diffuse functions as well as polarization, is needed to provide magnetic shielding results with 10% of the limiting values from very large basis sets. This conclusion of course also applies to the calculation of spin-rotation constants because of eq. (2). Our subsequent calculations on polyatomic molecules accordingly have employed basis (5). When computing spin-rotation constants via magnetic shielding constants we require the gauge origin for the up calculation to be at the nucleus of

UCHFlGA method Sadlej GA

~4b~~

CHF

method

1 2 3 4 5 BASIS SETS IN ORDER OF INCREASING SIZE

‘I &IS

+ --o--

Fig. 3. Paramagnetic shielding at F in HF versus basis set used.

interest (see eq. (2)) and therefore when approximate methods are used we must be satisfied that this is a suitable choice of gauge because, particularly for the GA method, the errors can be greatly increased for some gauge choices.

16

u L 2

10

3 +

UCHFlGA

+%

CHF

method

8

66 1

2

3

4

method

5

BASIS SETS IN OROER OF INCREAWG Fig. 4. Paramagnetic

shielding

6 SIZE

at H in HF versus basis set

R. D. Brown, M. Head-Gordon

Test calculations using various gauge origins showed that the CHF method with basis set (5) gives results almost gauge independent, as required, but that serious variations arise when the GA approximation is used. The SGA method performed noticeably better than the GA method but still was less satisfactory than the CHF calculations. We therefore turned to CHF calculations using basis set (5) to evaluate the quality of its performance for a collection of small molecules for which experimental data are available. Table 2 lists our results for eight molecules. The largest discrepancy is for the P nucleus in PN where U,, is calculated to be 15% greater than the experimental value. All other values for major components of U agree with experiment to better than 10%. We conclude that the CHF technique can give predictions of the U tensor with sufficient accuracy to provide helpful predictions for spectroscopists of the spin-rotation multiplets that they will encounter for small molecules. Although the feasibility of applying the CHF procedure and basis set (5) to molecules with more

/ Molecular spin -rotation

5

constants

than, say, four first-row atoms is doubtful with currently available computer facilities, it is probably only for such smaller molecules that spin-rotation tensors will be of interest because of the factors, especially (iii) below, determining the general magnitude of the interaction. The factors are: (i) A reasonably large nuclear g value, e.g., ‘IP, 19F ‘H and 14N. iii) A substantial paramagnetic shielding value at the nucleus of interest. This usually occurs for nuclei heavier than H. For first-row atoms it is typically negative and of magnitude several hundred ppm. (iii) At least one of the principal moments of inertia must be fairly small. This is necessary because of the l/I dependence of the interaction.

6. Conclusion For small molecules of spectroscopic interest it is feasible to compute ab initio values of the spin-rotation interaction tensor U with useful ac-

Table 2 SRCs calculated using the CHF method ‘) Molecule

Nucleus

CH,O

“0

Component

“C

N,O N2O

PN PN HCN HCN HCN co co CH,NH

-1.3 - 122.0

2 -127.9

(10) (10) (3)

1271

cc aa bb

- 20.5 - 5.3

-19.8 -7.7

(3) (3)

f281

P

aa bb, cc

end N middle N P N N “C H “C “0 N

bb, bb, bb, bb, bb, bb, bb, bb, bb, ; cc

CH,PH

“P

a) All omitted components are zero.

Ref.

371 29

cc PH3

M(obs) (kHz)

371.6 29.9

;; CH,O

M(calc) (kHz)

aa bb cc

cc cc cc cc cc cc cc cc cc

- 109.5 - 114.1 - 2.16 - 2.68 - 89.7 - 11.5 - 10.4 - 17.1 6.04 - 33.7 - 32.9 - 50.3 - 9.1 -0.5 - 254.4 -51.5 - 14.4

(10)

- 116.38 (32) - 114.90 (13) - 2.35 - 2.90 -78.2 -10.4 -10.4 -15.0 3.7 - 32.59 -30.4 -49.5 -9.7 -1.2 - 301.9 -50.1 -20.3

(20) (20) (5) (5) (3) (10) (3) (15) (12) (47)

1291

1301 [301 1311 1311 1321 ~321 1321 1331 1341

(27) (24)

151

(17) (7) (7)

1261

6

R.D. Brown. IU. Head-Gordon

curacy (normally better than 10% for major components of U) by using the CHF method and a moderately large bases set, such as (5). The use of the geometric approximation gives unreliable results. The use of the Sadlej approximation represents an improvement but was less satisfactory than the CHF results. References [l] G.C. Wick, Nuovo Cimento 10 (1933) 118. [2] W.H. Flygare, J. Chem. Phys. 41 (1964) 793. [3] R.M. Stevens, R.M. Pitzer and W.N. Lipscomb, J. Chem. Phys. 38 (1963) 550. [4] P.D. Godfrey and D. McNaughton, to be published. [S] R.D. Brown, P.D. Godfrey and K.P. Yamanouchi, to be published (thioformaldehyde). [6] J. Schulman and J. Musher, J. Chem. Phys. 49 (1968) 4845. [7] D. Tuan, Chem. Phys. Letters 7 (1970) 115. [a] D. Tuan and A. Davidz, J. Chem. Phys. 55 (1971) 1286. [9] A.T. Amos, J. Chem. Phys. 52 (1970) 603. [lo] A.T. Amos and J. Musher, Mol. Phys. 13 (1967) 509. [ll] A. Sadlej, Chem. Phys. Letters 58 (1978) 561. 1121 R.D. Brown, P.D. Godfrey and D.A. Winkler, Australian J. Chem. 35 (1982) 667. [13] R. Holler and H. Lischka, Mol. Phys. 41 (1980) 1017. [14] R. Holler and H. Lischka, Mol. Phys. 41 (1980) 1041. [15] F. Keil and R. Ahhichs, J. Chem. Phys. 71 (1979) 2671. [16] P. Lazzeretti and R. Zanasi, Intern. J. Quantum. Chem. 12 (1977) 93.

Molecular

spin - rotation

constants

1171 P. Lazzeretti and R. Zanasi, J. Chem. Phys. 68 (1978) 832. [181 P. Lazzeretti and R. Zanasi, J. Chem. Phys. 68 (1978) 1532. 1191 P. Lazzeretti and R. Zanasi, J. Chem. Phys. 72 (1980) 6768. PO1 R. Ditchfield, Mol. Phys. 27 (1974) 789. 80, version d, QCPE 446, Chemistry DepartVI GAUSSIAN ment, Indiana University, USA. WI W. Hehre, R. Stewart and J. Pople, J. Chem. Phys. 51 (1969) 2657. ~231 R. Ditchfield, W. Hehre and J. Pople, J. Chem. Phys. 54 (1971) 724. v41 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. 1251 R. Ahlrichs and P.R. Taylor, J. Chim. Phys. 78 (1981) 316G. unpublished data. WI P.D. Godfrey and D. McNaughton, 1271 W.H. Flygare and V.W. Weiss, J. Chem. Phys. 45 (1966) 2785. B. Krieger and J.S. Muenter, J. Chem. Phys. 1281 B. Fabricant, 67 (1977) 1576. R.M. Neumann, SC. Wofsy and W. v91 P.B. Davies, Klemperer, J. Chem. Phys. 55 (1971) 3564. [301 K. Casleton and S. Kukolich, J. Chem. Phys. 62 (1975) 2696. [311 J. Raymonda and W. Klemperer, J. Chem. Phys. 55 (1971) 232. J. Mol. Spectry. 50 ~321 R.M. Garvey and F.C. DeLucia, (1974) 38. [331 I. Ozier, L. Crapo and N.F. Ramsay, J. Chem. Phys. 49 (1968) 2314. [341 M.A. Frerking and W.D. Langer, J. Chem. Phys. 74 (1981) 6990.