Molecular constants of NSF3

JOURNAL

OF MOLECULAR

SPECTROSCOPY

Molecular

9,

107-113 (1962)

Constants

of NSF,

K. RAMASWAMY, K. SATHIANANDAN, AND FORREST F. CLEVELAND Spectroscopy

Laboratory,*

Department of Physics, Illinois Chicago 16, Illinois

Institute

of Technology,

Potential energy constants, rotational distortion constants, mean amplitudes of vibration, and Coriolis coupling constants for NSF3 have been calculated from the vibrational spectral data. Values of the heat content, free energy, entropy, and heat capacity also are given for the rigid rotor, harmonic oscillator approximation at 12 temperatures between 106°K and 1066°K. INTRODUCTION

The infrared spectrum of gaseous NSF3 was studied by Richert and Glemser (1) who proposed a Csu structure. Recently Kirchhoff and Wilson (2) analyzed the microwave spectrum for the gaseous state and confirmed this structure. NSF3 appears to be the only example reported to contain a sulfur-nitrogen triple bond. Three of the 6 normal modes of vibration are of type a and the other three are of type e. In the present investigation, potential energy constants were obtained for the general valence force potential function. Using these force constants, rotational distortion constants, and Coriolis coupling constants were evaluated. The meansquare-amplitudes of vibration were calculated by the secular equation method of Cyvin (3) and thermodynamic properties were computed with an IBM 1620 digital computer, for the rigid rotor, harmonic oscillator approximation. I. POTENTIAL In the present calculations, and Glemser

ENERGY

the vibrational

CONSTANTS wave numbers reported by Richert

(1) for the gas were used. They are: a-1515,775,

429, and 342 cycles/cm.

The equilibrium

and 521; e-811,

values of the internuclear

angles D, d, j3, and CY(D~= S-N

separations

= 1.416& d, = S-F = 1.552 A, = 122“22’) were obtained from Pe = F-S-F = 94”2’, and 01~= N-S-F microwave data (2). The symmetry coordinates, the potential function, and the F and G matrices used were the same as those of Ziomek and Piotrowski (4)) except for the following changes. Due to the large deviation from tetrahedral and interbond

* Publication No. 164. 107

108

RAMASWAMY, SATHIANAKDAN,

BND CLEVELAND

angles, the type a symmetry coordinates Rs and Rd were modified to Ra = (3 + 3~‘) -%m

+ Acuns+ Aala + Y ( N% + AL%+ 4%) I

Rr = ( 3 + 37’) -“*[Aw

+ Aarzs+ AW - Y ( M

and +

432

+

AL%)

I,

where y = 3l” cos p/cos &Y, in order to satisfy the redundancy condit’ion The modified element’s of the G matrix for the a t’ype are Gk = -

3PLs (3 + 3r2)l’?

2cosp(1 - cosa) d sin OL

G

V&S (3 + 3y2)l’”

2(1 + 2 cos a)(1 d sin a!

= -

+

,y sG

(L4

+Bcosp)

1

(5).

,

cos a)

1lj

3A cos P + B(1 + 2 cos cy) -~ sin p i

+r and 3PF Gn3R = (3 + 3r2)

2(1 + c;

+&

?I

4 cos2 a: + ;; + 27 _ J2 cos p( 1 - cos Ly) sin a! sin p d2 \

3W-Y2 [I + 2 cos Ly+ (3 + 3y2)D2 sin” fl 3l.k + (3 + 3-P) + &@

3 COGfl]

4(1 + 2 cos cr)(l - cos a,)2 d2 sin2 LY

{3A2 + B2(1 + 2 cos cy) + GAB cos p)

+ ‘;(s!n-;;

;)

1.

(3A cos p + B(1 + 2 cos Ly))

For type c, the only modified element is G;z = g + where

P,

(’ ;,;;a)B’+$+y), and

B=(&-y).

Corresponding changes were made in the F matrix. Using these, the potential energy constants were evaluated and the results are given in Table I. It may be pointed out that it, was not possible to adjust the equations when the Urey-Bradley potential fun&ion was used.

MOLECULAR

CONSTANTS TABLE

I

POTENTIALENERGY CONSTANTSIN md/A F-matrix elements

12.3WQf 6.9700 -1.2000 6.0000 0.3000 5.7830

Bond stretching and bond-bond interaction constants jn

12.3OOt 4.7767t 0.5600 0.6117

fd

FOR

NSF3 Bond-bond interaction constants

Angle bending and angle-angle interaction constants ja js k;

0.6822 0.9030 0.0302 0.4412 -0.4869 -0.2821

f P@ f afl

0.2600 0.3000 1.1000 0.4509 1.4500

109

OF NSF,

j& fDs f$ fda

.fdP f&

-0.2899 0.3942 0.0327 0.2306 0.0525 -0.1814

* This number of significant figures is retained to give best reproduction of the observed wave numbers and to secure internal consistency in the calculations. t Values of 12.4 and 5.0 md/A were reported in Ref. 1 for jo and jd, respectively. II. ROTATIONAL

DISTORTION

CONSTANTS

The expressions for the rotational distortion constants, DJ , DJK , and Dg for axially symmetric CXsY type molecules were given by Kivelson and Wilson (6). These constants were calculated for NSF3 by the method given by Dowling et al. (7). The elements [JzzR3]0and ]JzlR310 were multiplied by the factor -3-l” cos ?&Y/COS /3 (4) to account for the changes made in the normal coordinate treatment,. The values obtained are: D, = 0.190 Kc/set;

DJK = 1.431 Kc/set;

and

DK = - 0.634 Kc/see. III.

MEAN-SQUARE-AMPLITUDES

In order to calculate the mean-square-amplitudes of the vibrations, the secular equation method of Cyvin (3) was used. The elements of the mean-square amplitude matrix z were obtained by solving the secular equation ]I;F-EA3L.\=O,

(I)

where F is the potential energy matrix, the values of A are connected with the normal frequencies Q. by the relation Ak =

(h/8&)

coth (hv,/2kT)

and xs = 4rr2C2Vk2.

110

RAMASWAMY,

HATHIANANDAN,

AND CLEVELAND

Here h is Planck’s ronstant, lo is Bolt,zmann’s constant, T is t,he absolute temperature, and c is the velocity of light,. The symmetrized matrix 2 contains the mean-square-amplitude quantities (,m) of the internal coordinates and the nonbonded distances. The relations connecting u and Z: can be obtained as follows: A set of internal coordinates and the nonbonded distances (r) can be related to the set, of internal coordinat#es (q) by t,he expression r = Aq,

(2)

where A can be obtained from the geometry of the molecule. If r is related to the set of symmetry coordinates S by

t,hen

r = VS,

(3)

V = AU’.

(4)

Here U’ is the transpose of the orthogonal matrix of transformation from internal coordinates to symmetry coordinates. The elements of matrix u (*an be obtained from u =

VZV’.

(5)

Of all the elements of the u matrix, the import’ant ones are upI and uq , where CJd =

(d2),

at3= D4P2),

UD

=

(D2),

UP = (?a

cd

, uD

, ua

, ‘~0 ,

ua = d2(a2), and

gq = (q2).

Here p and q represent the nonbonded FE’ and NF distances, respectively. relations connecting these quantities with the 2 matrix are:

The

MOLECULAR

CONSTANTS TABLE

OF NSF,

II

MEAN-SQUARE-AMPLITUDES OF NSF,

AT 298.16”K

Symbol

Value in A2 0.001200~ 0.001933 0.001015 0.001673 0.002592 0.003392

CD

‘Td =. QB ap 4 * This number calculations.

111

of significant

figures is retained

to secure

internal

consistency

in the

and ~4 = Ao%

2AoBo

+

v3

2AoCor

a

812 +

(3 + 3y9”2 C02T2 +

(3 + 3-d

2BoCoy

zz +

&?(3

+ 3y2p2 zi3

Z;, + ; &%I

+ ;

BoCoZI2

+

;

Co%

where Ao = (D -

Bo = (d -

dcosP)/q,

Do = d(l -

cos ar)/p,

DcosP)/q,

Co = Dd sin p/q,

E, = d” sin q/p.

and

The results are given in Table II. IV. CORIOLIS

COUPLING

CONSTANTS

The relations connecting {, F, and G matrices were given by Meal and Polo (8) ; they are L’FCL’-’

= A<

-

= n(E

(6)

and L’F(G

C)L’-’

-

0,

(7)

where F and G correspond to the degenerate type e, L is the matrix of transformation obtained from ) FG -

En 1L = 0,

(8)

< is the matrix containing the zeta elements, C is a symmetric matrix which depends only on the mass and geometry of the molecule and which can be obtained by the relation CYj = C /-b(% X Sj.) ‘em a

(9)

,

112

RAMASWAMY,

SATHIANANDAN,

AND

CLEVELAND

where a represents the X, Y, or Z axes, Si, is Wilson’s S-vector for the internal coordinate i and the atom a, and e, is t,he unit vector along cy. Using Eq. (9)) the elements of the C matrix, for the type of molecule under consideration, were obtained. They are

sin (Ycos s/sin’ p + v’?&~ sin a cos 6/D’ sin’ 0

-

&pa

(A- y)

(1 -

1,

cos a) cos 6/d sin 01,

and CL,% = [ - dfjpclF( 1 + 2 cos a) cos 6/d’ sin a! +

&&(

1 -

cos a)” cos 6/d’ sin (~1.

Here p is the reciprocal of the mass of the atom, and 6 is the angle between t,he S-N bond and the normal to the F-S-Fplane. TABLE

III

HEAT CONTENT H, FREE ENERGY F, ENTROPY S, ANU HEAT CAPACITY C, IN CAL DEG-1 MOLE-’ FOR NSF3 AT ONE ATMOSPHERE PRESSURE AND FOR THE IDEAL GASEOUS STATE

T(“K) 100.00 200.00 273.16 298.16 300.00 400.00 500.00 600.00 700.09 800.00 900.00 1000.00

(HO -

&‘)/T

8.15 9.74 11.17 11.64 11.67 13.38 14.79 15.95 16.91 17.71 18.38 18.96

-(P

-

E”O)/T

47.30 53.38 56.62 57.62 57.69 61.29 64.43 67.24 69.77 72.08 74.21 76.17

so 55.45 63.11 67.79 69.26 69.37 74.67 79.22 83.19 86.68 89.79 92.59 95.13

Go 9.05 13.63 16.40 17.17 17.23 19.60 21.17 22.25 23.01 23.56 23.97 24.29

MOLECULAR CONSTANTS OF NSF,

113

From Eqs. (6) and (7)) it is seen that trace j FC j = c

&ci

and trace 1F(G -

C) I = Cad1 - Si).

(10)

For the symmetric rotor molecules, the sum of the &‘s is equal to 1,/21, , (9)) where I, is the moment of inertia about the Z-axis apd I, is the moment of inertia about the X- or Y-axes. A suitable set of { values which conforms to the relations given above was ohtrained. The values are lie = 0.724, V.

lz” = - 0..502,

and

[se = 0.101.

THERMODYNAMIC PROPERTIES

From the previously given wave numbers for the gaseous state, the heat content, free energy, entropy, and heat capacity were computed by the usual statistical methods with an IBM 1620 digital computer. The calculations were made for one atmosphere pressure for a rigid rotor, harmonic oscillator approximation. The symmetry number is 3, and the moments of inertia are I, = 103.114 and Ig = 1.59.643 amu ;i”. The values thus obtained are given in Table III. ACKNOWLEDGMENTS This investigation was a part of a research program which has been aided by grants from the National Science Foundation. The authors are grateful for this assistance. RECEIVED:

February 27, 1962 REFERENCES

1. H. RICHERT AND 0. GLEMSER, Z. anorg. IL. allgem. Chem. 307, 328 (1961). 2. W. H. KIRCHHOFF AND E. B. WILSON, JR. (private communication, September 25, 1961). S. S. J. CYVIN, Acta Chem. Stand. 13, 2135 (1959). 4. J. S. ZIOMEK AND E. A. PIOTROWSKI, J. Chem. Phys. 34, 1087 (1961). 5. J. M. DOWLING, R. GOLD, AND A. G. MEISTER, J. Mol. Spectroscopy 2, 411 (1958). 6. D. KIVELSON AND E. B. WILSON, JR. J. Chem. Phys. 20, 1575 (1952); 21, 1229 (1953). 7. J. M. DOWLING, R. GOLD, AND A. G. MEISTER, J. Mol. Spectroscopy 1, 265 (1957). 8. J. H. MEAL AND S. R. POLO, J. Chem. Phys. 24, 1119 (1956); 24, 1126 (1956). 9. R. C. LORD AND R. E. MERRIFIELD, J. Chem. Phys. 20, 1348 (1952).