Atomic force constants and apparent electron densities in elemental sulfur

Speotrochimica Acts, Vol. 31A, PP.1421 to 1425. Pergamon Press,1975. Printedin Northern Ireland

Atomic force constants and apparent electron densities in elemental sulfur w.

T. &NQ

Metcalf Researoh Laboratory, Brown University, Providence, Rhode Island 02912, U.S.A. (Received24 August 1974) &&a&-The atomio foroe constants for sulfur in the ring compounds S,, S, and S,, are found to be 476 f 2 N/m, and to be transferable between these three sulfur compounds. Using a simple “atoms in molecules” model to approximate the electron density for a sulfur atom, the magnitudes of the force constant for sulfur was computed and found to be consistent with the structures of these compounds.

INTRODUCTION In a recent paper, STEUDEL and EGGERS [l, 21 reported the vibrational

assignments and force con-

stants in S,,, and compared their results with those found by SCOTT, MCCULLOUGH and KRUSE [3, 41 from a similar analysis of S,.

tion of the generalized harmonio valence force constants usually considered in a vibrational analysis [S]. Atomic force constants are related to the normal modes of molecules by the frequency sum rule, originally derived by DECIUS and WILSON [13], but expressed in a Cartesian coordinate representation [8, 141,

Earlier, a comparable

analysis of the spectra of S, was reported by NIMON and NEW

[5, 61, who also compared their results

with the force constants in S,.

It was found that

the normal modes in these ring compounds accounted potential eters.

for by a modified Urey-Eradley function

containing

Consequently,

were model

only a few psram-

it was suggested

that

this

model is generally applicable to sulfur ring systems, and that the small, but significant differences in some of the model parameters in the different systems were explicable in terms of the interactions between lone pair electrons in these ring structures. In this paper the similarity between S,, S, and S,, is also examined,

but from the different point

of view provided by the so-called atomic force constants [7, 81. These force constants constitute one important

property

of molecular

potential

I

in which the wj are the normal harmonic frequencies (0, = 27rcovj), and the prrare the reciprocals of atomic masses. The summations extend over all normal modes, j, and over all atoms, a, respectively. Although an atomic force constant describes only one property of the total molecular potential function, that property is of considerable interest for several reasons. Of primary importance here, the sum rule (2) relating these force constants to molecular frequencies does not depend explicitly upon molecular geometry. This makes it possible to compare the atomic force constants for an atom in different molecules directly, without introducing additional assumptions about the effects of redundancies and other constraints on the force constants or about the specific molecular force field. For example, since the sulfur atoms in S, and S, are all symmetrically equivalent [3-61, the sum over c(in (2) can be reduced to a single term,

func-

2 wje = ,usNsVsaU

tions that is readily determined for these systems, without

introducing

models.

It is found that the atomic force constant

specific

potential

function

for sulfur is essentially identical in these three ring systems, and that their magnitudes

are accounted

by a simple “atoms in molecules” model for atomic force constants applied to these structures [B-12]. THEORY The atomic force constant, V,*lJ, for an atom ccin a molecule, is defined as the Laplacian of the molecular potential energy function U(R), differentiated with respect to the Cartesian coordinates of the atom considered, V,=U =

awlax, + awlay,= +

aw/a2,8 (1)

and is nothing more than a particular linear combina-

a-

(3)

in which Ns is the number of sulfur atoms in the ring. Consequently, the atomic force constants for sulfur in S, or S, are found from (3) simply by computing the appropriate squared frequency sums. Although the sulfur atoms in S,, are not all exactly equivalent, equation 3, with N, = 12, is used here to determine an averaged value for VsaU in S,,, and, based upon the theoretical analysis outlined below, it is anticipated that the dispersion of this averaged value is quite small. A second reason for the interest in atomic force constants is that they are simply related to the electron densities in molecules. It has been shown that [lO-121

v,‘u

= ~-%p(R,) - %(v,p - V,(l/lr

- R,])) (4)

where p(R,) denotes the total electron density at nucleus a, and 2, is the nuclear charge. If the electron

1421

W. T. Kma

1422

CALCULATIONS AND RESULTS

density is represented by an expansion in atomic orbitals centered on each nucleus [15] p(r) =c

x C/VXS&

B>Y

-

WXYA

-

BYI

(6)

i.j

in which, for instance, xpi(v --Rg) denotes an orbital on nucleus /?in quantum state i, and Cp,o is an appropriate density matrix element, then (4) can be reduced to the sum of three terms [9, 10, 121, v,2u

= 471.%[ 2 ps(%) Bita

+ c

analysis

atomic

force

termined

of the sulfur ring systems,

constants

cies for these systems cited

above,

using

@@rW (6)

~a(%)

= c C~‘%i(R&pr(%)

Qgy(R,) = --@a) 2 k;i IV&lb i.j

(7)

- R,I)I xrj * v,Cpr*9 (8)

and (9) i.’

It should be pointed out here that Equation 6 is essentially exact, in that only theBorn-Oppenheimer approximation and a few mathematioal approximations concerned with the expansion in (6) have been used in its derivation. The first term represents the interaction of the nuclear charge on 0: with the charge distributions, pb[R,], centered on the other atoms in the molecule, the second term represents the interaations resulting from the redistribution of charge due to nuclear displacement, and the third term represents the interaction of the nuclear charge with the non-classical “interference” electron density [15] in the moleoule. According to Equation 6, then, atomio force oonstants depend upon properties of the total electron density in molecules. Because of this it is possible to reduce this equation to a more tractable approximate form and to obtain a useful structural interpretation of atomic force constants. To this end, a very simple form of the “atoms in molecules” model [16] is adopted whereby the terms (8) and (9) are assumed to be negligible, so that (6) is reduced to a linear superposition of two center terms called “bond constants,” V,‘u where K,,M,p)

N c K&W ##Cc

(16)

= 4n-%p,M,)

The approximations by which (6) is reduced to (10) are based upon the general observation that total molecular electron densities are very nearly equal to those obtained by superimposing the densities of the constituent atoms, and they are justified in detail, at least for first and second row elements, elsewhere [g-12]. A third reason for the interest in atomic force constants that is not directly related to the goals of this paper, is nonetheless mentioned in passing here. These force constants play a direct role in determining heat capacities of materials and the effects of isotopic substitution on the kinetic and thermal properties of materials, particularly at high temperatures [9, 17-191. Consequently, their characterization through (2) or (6) is of considerable interest in establishing the relationship between the structure of a material and its properties.

estimated

and

Second

if they are consis-

the electron by

density

in sulfur is

computing

in hydrogen

the

atomic

sulfide.

The sums of the squares of the normal cies for these sulfur compounds

frequen-

are listed in Table

1, and the resulting atomic force constants and S,,, given by Equation 2 [22].

stants

in S,, S,

3 (Ms32 = 31.97207 u21)

are listed in the column headed Table

the

are estimated,

of these sulfur compounds.

tested

force constants

are defrequen-

in the references 3 [20].

10, to determine

tent with the structures For this purpose

where

reported

of these force constants

using Equation J_

Vs2U,

fundamental

Equation

S,, S,

First, empirical

for sulfur,

from the observed

magnitudes

%y(R,)

B,Y +

The

and S,,, is carried out in two steps.

vs2U(observed)

in

It is found that the atomic force con-

are essentially

equal

and, therefore,

trans-

ferable between

all three ring systems.

test, a common

value for Vs2U in these compounds

was determined the resulting

by least squares adjustment,

value

shown in Table The

squared

averaged

frequency

with

is given

and is found

(S),

are

average.

using

in Table

to differ

this 1 as

from

the

sums by less than one percent in all cases.

addition

to these

force constants

ring

in hydrogen

systems,

H,S

and

D,S

sums are listed

[23J.

frequencies

The

in Table

tlie

atomic

sulfide were also com-

puted using the fundamental for

and

its dispersion

sums derived

constant

z:y2(calculated), In

along

2 in the row labelled

force

observed

As a further

squared

reported frequency

1 and the atomic

force

constants derived from, using (3) (Mn

= 1.00783 u,

M,

Table

= 2.01410 uzl),

Vs2U(observed) are used to test constants,

are

the analysis

proceeding

implications

considered,

estimated.

expression

densities

the proper-

the relationship

(10).

of the

using the approximate For this purpose,

in sulfur and hydrogen

the

need be

One rough estimate that has been found

to be adequate

in most cases is the approximation

using nodeless Slater orbitals Q(R,)

of any of the

their structures and the magnitudes

theoretical

as

below.

of these results regarding

force constants is examined,

2

These data

of the srrlfur force

to a discussion

ties of the compounds

electron

in

and will be considered

Before

between

listed

and VH2U(observed).

[9, lo],

N (Ne/~)[(268)2”+1/(2n)!](R,g/a,)2(”-” x exp

in which afl = 2,*/n

(--2~~-&la,J

(11)

Atomic force constants and apparent eleatron densities in elemental sulfur

1423

Table 1. The squared frequency sums for elemental sulfur and the hydrogen sulfides in kilo Kayser units (kK)a.

Iv2

cv2a

(ohs)

b S.5 Sa

c

S12

d

Error

(talc)

%

1.517

1.513

-0.3

2.035

2.017

-0.9

3.009

3.026

0.6

H*S e

15.134

c2s e

7.887

.a The calculated values derived from averaged value of V S% b

Ref. (5); =

Ref. (3); d

Ref. (1); e

see text;

Ref. (23).

Table 2. Observed and calculated atomic force constants, in N/M elemental sulfur and hydrogen sulfide.

(calc.1) a

Cobs)

(calc.1) a

(calc.11) b

479

Ss

472

S12 (S)*W

475k2 1184

H2S

b

(ohs)

476

s6

a

(calc.11) b

units, for

574(20.8)'= 1317(11.2)

46S(-1.5) 1186(0.2)

431

472(9.5)

444(3.0)

Slater charges for S and H are 5.45 and l.OO,respectively, ref. (25); Adjusted Slater charges for S and Hare

text; '

5.60 and 1.05, respectively, see

Vercent error shown in parentheses.

where VI is the principle quantum

number of the

Only the valence shell need be considered in (ll),

valence shell of atom /?, e the electronic charge, N

for the inner shell, core electrons make negligible

the number of valence electrons and a, the Bohr radius. The parameter Zs * is a screened nuclear

lengths,

charge.

neighbor atoms need be considered in (10).

This approximation

assumes that the elec-

contributions

to pS(R,)at normal covalent bond Rap. For the same reasons, only nearest

tron density of an atom in a molecule is essentially

Using standard Slater charges, 1.00 for Hydrogen

a spherical one, in which all orbitals in the valence

1s orbitals and 5.45 for sulfur third shell orbitals

shell are described by the same screening constant

[25], the atomic force constants were calculated and

a.nd are equally occupied by the valence electrons.

are shown in the columns

headed Caloulation

1.

W. T. KING

1424 In these calculations an averaged S-S of 0.2060 nm [l], and R,,

bond length

= 0.1328 nm [26] were

heat capacities per degree of freedom, is also equal at high temperatures,

used in ( 11). Considering the rather crude estimates

systems,

of the electron densities in these atoms, the results

molecular geometry [Q].

of this calculation are quite good.

values of the atomic

Consequently,

Finally, it is found that the atomic force constant for sulfur is not transferable between all sulfur com-

for hydrogen,

charges

2, Calculation

to (lo),

was carried out

of 5.60 for sulfur and 1.05

and the resulting

given in Table According

force constants.

a second calculation

using adjusted

force constants

are

2.

the magnitudes

sulfur ring compounds

of Vs2U in the

and VH2U in hydrogen

sul-

pounds.

Unlike

the atomic presumably

good agreement

of the derived

both of these force constants, adjustment validity

of Z,*

obtained

in calculation

by a small

2 supports

the

the constant

depends

Vs2U in hydrogen

upon the magnitude

small adjustment

of Zn*,

in its magnitude

good agreement

analysis

the atomic

is required

a to

with experiment.

indicates

that

force constants

deed consequences

the magnitudes

other

of their similarity

interatomic

sulfur

atoms

distances,

of

in S,, S, and S,, are inin structure.

In these systems each sulfur atom is coordinated two

for sulfur in (S),

different

in other sulfur compounds

in these compounds,

and analogous

compounds

second

by Equation

row

Be-

of Vs2U

differences

in

elements

[Q], is

in molecules”

model

10, it is suggested

is a useful, though

of molecular

as well.

the values

for by the “atoms

this model analysis

of other

com-

from that in H,S and

potential

limited

that

one for the

functions

of com-

pounds of first and second row elements.

sulfide,

and only

DIsCUSSION This

in com-

of (10) applied to these two force constants.

Similarly,

obtain

values of

constant

between

accounted

The very

observed

cause the differences

summarized

bond distances.

force

pounds is distinctly

at the appropriate

or H-S

the behavior

pounds of first-row elements [7], the magnitude of

fide both depend upon the electron density in sulfur S-S

of their

the screening

Zg* in (11) to better reproduce the ex-

perimental

independent

The errors can

be reduced, of course, by adjusting constants

and is, therefore,

C,/(3%6),

for all three

at very

nearly

0.2060 nm [l].

to

the same

According

to

(lo), then, the atomic force constants, even for S,,, in which the atoms are not all equivalent,

should

all be equal, as observed. There

is, apparently,

bond length

tions for thermal

motion

diffraction

data.

a corrected

S-S

ment

the bond

with

a question

in S, arising from

about the S-S

the proper

in the analysis

correc-

of X-ray

CARON and DONOHUE [27] report bond length of 0.2060 nm in agreelengths

reported

for S, and

S i2. ABRAEAMS [ZS] on the other hand, states that 0.2045 nm is the correct value for this bond length. Using this shorter bond length in (11) yields a value of 507NJm

for Vs2U,

than that computed

a value

about

at 0.2060 nm.

differences

in the observed

are found,

it is suggested

bond length reported

atomic

6% greater

Because no such force constants

that the corrected

by Caron and Donohue

S-S is the

more accurate. Another

similarity

fur compounds cause the atomic three systems,

in the properties

is implied force

by these

constants

the vibrational

of these sulresults.

Be-

are equal in all

contribution

to the

REFERENCES [I] R. STEUDEL and D. F. BQaERS, JR., Spectrochim. Acta 31A, No. 7 (1976). [Z] R. STEUDEL and M. REBSCH, J. Mol. Spectry. 51, 189 (1974). [3] D. W. SCOTT, J. P. McCaLoUaH and F. H. K&USE, J. Mol. &m&y. 13, 313 (1964). [4] D. W. SCOTT and J. P. MCCULLOULXX,J. Mol. &e&y. 6, 372 (1961). [a L. A. NIMON and V. D. NEFF, J. Mol. Spectry. 26, 175 (1968). [61 L. A. NIMON, V. D. NEFF, R. E. CANTLEY and R. 0. BUTTLAR, J. Mol. Spectry. 22, 105 (1967). 171 R. R. GAUQHAN and W. T. KINQ, J. Chem. Phys. 57,463O (1972). PI W. T. KINQ aud A. J. ZELANO, J. Chem. Phys. 47, 3197 (1967). [91 W. T. KINQ, J. Chem. Phys. J. Chem. Phys. 61, 4026 (1974). [lOI W. T. KING, J. Chem. Phys. 57, 4636 (1972). PII A. B. ANDERSON and R. G. PARR, Theor& China. Acta (Bed.) 25, 1 (1972). WI A. B.‘ AN&R&IN, j. C&n. Phys. 58, 381 (1973). u31 E. B. WILSON, JR.. J. C. DECIUS and P. C. CROSS, Molecular V&at&, McGraw-Hill, New York, (1965). P41 J. BIEQLEISEN, J. Chem. Phys. 28, 694 (1958). WI See, for instance, K. RUDENBERQ, Rev. Mod. Phyu. 34, 326 (1962). [I61 See, in particular, T. ARAI, Rev. Mod. PhyS. 32, 370 (1960). P71 W. T. KINQ, J. Phya. Ohem. 77,277O (1973). WI J. BIEQELEISEN and M. WOLFSBERQ,Advarb. Chem. Phya. 1, 15 (1958). [I91 J. BIEQELEISEN, J. Chem. Phys. 23, 2264 (1955). WI Calculated values for the unobserved A,, and A,, modes for S,, in Ref. [I] were used to compute the sum in the Equation 3. WI A. H. WAPSTRA and N. B. GOVE, Nuclear Data Tables, AQ, 265 (1971).

Atomic force constants and apparent electron densities in elemental sulfur [22] The International System of Units is used throughout this paper. Force constants are expressed in Newtons per meter (N/m), for this unit conveniently places the decimal point just to the left of insignificant figures in most cases. Force oonstants in millidyne/angstrom units are obtained by dividing these results by 100. [23] T. S~XQ?OUCHI, editor, N&l. St. Ref. Data Ser. N&Z. Bur. of Std. (U.S.) 39 (1972). There seems to be an error in the value reported for vQin D,S here. Using the produot rule and the other fundamentals listed, the value of vs is 1887 K instead of 1999 K as reported. The value consistent with the produot rule is used here.

21

1425

[24] J. A. POPLE end G. A. SEQAL,J. Chem. Phys. 43, 5136 (1962). [25] J. A. POPLE and D. L. BEVEBID~E, Approsimate Molecular Orb&al Theory, McGraw-Hill, N.Y., (1970) [26] L. E. Srrrroa, editor, lnlerotomti Distances Spec. Publ. No. 11 andNo. 18. The Chem. Sot. (London), (1968). [27] A. CAROWand J. DONOEUE, Aok Cryst. 18, 662 (1965). [28] S. C. ABRAHAMS,Aota. @pt. 18, 566 (1966).