The force field of phosphorus

Speakochimka Acta,Vol.%‘A,pp.778to776.Pergamon Press 1971.Printed InNorthern Ireland

The force field of phosphorus* ROBIN S. MCDOWELL Los Alemos Scicntih Laboratmy, University of Celifomia, Los Alamoe, New Mexico 87644 (Received21 Azcgclet1970) &&&-The infraredspectrum of P, in CS, solutionswas obtained for peth lengthsup to 6 mm. A breddown in the T, selectionrules due to intermolecularinteractionallowed al1fundamentals and binary combinationsto be observed in the infrared. The resultinga+nharmonicity const&nta wereusedto estimatethe zero-orderfrequencies,ad from these the force field and mean tEmplitudes of vibration were cctlculated. The general quadratic valence force constants aref? = 2.19, f, = -O.l3,f,,’ = 0.08, all in mdyn/A. INTRODUCTION

tetrahedral P, molecules (symmetry Td) in the vapor, liquid and solid (white phosphorus) states [I] ; the P-P bond length as determined by electron diffraction in the vapor is 2.21 f 0.02 A [2]. For a molecule of this type there are three vibrational fundamentals with distribution A, + E + F,. Since there is only one fundamental of each symmetry type, the calculation of the general quedratic force constants is trivial once the zero-order frequencies have been established. To date, however, this hrtsnot been done, and the only reported force constants have been based on the observed frequencies [3], or on the observed frequencies with partial anharmonicity corrections of doubtful validity [4]. These two treatments were in fair agreement but differed in the sign of one of the interaction constants. Previous spectroscopic work on phosphorus includes early Raman studies on all three ph=es and on CS, solutions [S], and a recent Raman spectrum of the vapor [6]. The infrared spectrum has been reported for the vapor [7], CS, solutions [7, 81, and the solid [7]. These studies established the fundamental frequencies, which for CS, solutions are 604 cm-l (Ye,A,), 361 cm-l (Q, E) and 462 cm-l (Ye,F,). However, the reported frequencies were not always in close agreement, and this, together with the fact that only a few overtones and combinations were observed, precluded obtaining a dependable set of anharmonicity constants. In the present study of P, solutions, the frequencies of all fundamentals and binary combinations were carefully determined. The resulting anharmonicity constants were used to determine the zero-order frequencies and from these the general quadratic force constants. PHOSPHORUSexists as

* This work was sponsoredby the U.S. Atomic Energy Commission. [l] J. R. VAX WAZER, Phoc+ph and Its Camped, Vol. 1, pp. 96, 102f, 107. Interscience (1968). [Z] L. R. I&~~ELL,S. B. HENDRICXSand V. M. MOSLEY,J. Chm. Phys. 8,099 (1936). [3] N. B. SUTER, Tram. Faraday Sot. 50, 207 (1964). [4] C. W. F. T. PISTORIUS, J. Chem. P&8.99,1421 (1958). [6] C. S. VENKATESWARAN, Proc. I&&z% Acad. sci. A$&260 (1935); A4, 346 (1936). [S] G. A. OZIN, Chsm. Commun. 1326 (1969). [7] H. S. GUTOWSKYand C. J. Holr~. J. Am. Chsm. 6% 7s, 6761 (1950). [S] H. J. BER~~IN and J. Powz~~a, J. C&m. Plbys. l&l018 (1960). 773

714

RoBIN

8. MCDOWELL

EXPER~ENTAL The spectrum of white phosphorus dissolved in CS, was obtained with Perk&Elmer Model 621 and Beckman Model IR-11 spectrometers out to a long-wavelength limit of 220 cm-r. One mm CsBr and 6 mm KBr cells were used, and the concentrations varied from dilute to nearly saturated ( -90 g phosphorus/ 100 g CS,) . Several of the P, bands occur on the shoulders of more intense CS, bands, and accordingly particular attention wae paid to background corrections: each band was examined both with a compensating solvent cell in the reference beam, and uncompensated, with the solvent absorption determined separately and subtracted from the sample spectrum. The spectrometers were calibrated using the IUPAC tables [Q] out to 690 cm-l and the rotational spectrum of water vapor [lo] at longer wavelengths. The observed spectra are summarized in Table 1. The positions of all bands were well determined with the exception of 2vr: this extremely weak peak almost coincides with a weak CS, peak at 1198 cm-l, and its exact position was somewhat uncertain. Table 1. Infrared spectrum of P, solutions Frequency

Assignment

1200 f 6 (vvw)* 1170 f 1 (vvw) 1113 & 1 (w)t 1061.7 (w) 996.3 (vw)? 966.6 (VW)* 917.9 (w) 816.3 (ww) 720.7 (ww)* 604.4 (mw)* 461.6 (s) 361 f 1 (mw)*

2% 235 + VII '*P,) + v1w,1 Vl + % YlP4) + v&S,) Vl + %I 2% % +%I 2% Vl 3 v2

Band positions are accurate to f0.6 cm-r except as noted. * Bands inactive for T, symmetry. t Intermolecular combination bands. Liquid CS,: v1 = 666 cm-l, vs = 392 cm-l. DISCUSSION

Two remarkable features of the solution spectrum are the presence of bands normally inactive under T, symmetry, and the presence of intermoleoulsr combination bands. Both of these are to be attributed to intermolecular interaction. Selection rule violations in the infrared spectrum of phosphorus were fist noted by GUTOWSKY and HOFFMAN [7], who reported the presence of y1 in solutions. However, they could find no trace of any forbidden transitions in the spectrum of the vapor end concluded that the spectra are consistent with tetrahedral symmetry, but with strong intermolecular forces operative in the solution and solid. This breakdown in the selection rules is very convenient for the purposes of the present work, since otherwise it would have been difficult or impossible to obtain the full set of anhitrmonicity constants. (All fundamentals and combinations are, of course, Raman [Q] IUPAC Commission on Molecular Structure and Spectroscopy, Tab&a of Waw-numbem for the Calibratkm of Infrared Spectrometers. Butterworths (1961). [lo] K. N. RAO, W. W. BRIM, V. L. SINNETTand R. H. WILSON, J. Opt. Sot. Am. 52,862 (1962).

The force field of phosphorus

776

active, but the Raman spectrum of a concentrated solution of P, in CS, at high gain showed only the P, and CS, fundamentals ; the combinations and overtones must be very weak in the Raman effect.) Intermolecular combination bands involve a simultaneous transition to the excited vibrational states of two different molecules. They were first observed in compressed gas mixtures [ll], and later in CS, solutions of several compounds [12], by KETELAARet al. (who termed them “simultaneous vibrational transitions”). Two such bands involving two different CS, fundamentals. were observed in the present work. The energy levels of a tetrahedral P, molecule, relative to the ground vibrational state, can be written

where the w( are the vibrational quantum numbers, the d, are the degeneracies, and IV is a function of the angular momentum quantum numbers [13] which can be neglected for our purposes. Fitting equation (1) to the observed spectrum of Table 1 yields the zero-order frequencies o1 and anharmonicity constants X,, of Table 2(a, b) ; Table 2.

Vibrational

constants of P, in solution

(a) Zero-order frequenoies (am-l) w1 = 619 f 6 w, = 313 l 4 co, = 481 f 8 (b) Anharmonicity constants (cm-‘) i=l

%

-4.4

f

i=2

2.6

+0.2 f -0.7 f

& X,, (0) Form

i=3 1.2 1.0

-4.2 -6.7 -2.6

f f f

0.9 0.9 0.6

oonstant.3

f, = 2.19 f 0.02 mdyn/A f, = -0.13 f 0.01 mdyn/A f,.,’ = 0.08 f 0.02 mdyn/A (d) Mean amplitudes of vibration

u(P-P)

kr = 1.76 f 0.03 mdyn/A Ea = 0.37 f 0.04 mdyn.A/rad* IC,ZD~ = -0.08 f 0.02 mdyn.A/rada

(A)

T=O’K

0.0469 f

0.0001

T = 298OK 0.0620 f

0.0002

the quoted uncertainties in these constants reflect the estimated accuracy of the observed vibrational frequencies. The most general quadratic potential function for P, can be written

the relations between the force constants and the frequencies are available in the literature [3, 41. The force constants obtained from the zero-order frequencies are [ll] J. FAHRE~ORT and J. A. A. KETELAAR, J. Chem. phy8.22, 1631 (1954). [12] J. A. A. KETELAAR and F. N. HOOQE, J. Chem. Phys.23,749,1549 (1955). [13] K. T. HECFIT, J. Mol. Spctry 5, 355 (1960).

776

ROBIN 8. MCDOWEU

given in Table 2(c) together with their estimated uncertainties. These force constants are quite close to those reported by SMTER [3], who used the observed frequencies in his calculations. They agree somewhat less well with the partially anharmonicity-corrected values of PISTORIUS[4], and in particular do not support his finding of a negative value for the interaction constantf,‘. Pistorius’ anharmonicity corrections, however, are highly suspect: they imply or - rr = -2 cm-l and 03 - % = 42 cm-l; the present values of 15 cm-l and 20 cm-l, respectively, seem rather more reasonable. Equation (2) differs from the usual polyatomic force fields in containing no bending force constants. A potential field which does can easily be constructed by a different choice of coordinates ; for example, (3) (A less general form of this equation, with k,, = 0, was used for phosphorus by BHAGAVANTAM and VENEATARAYUDU [14].) This alternative set of force constants is also given in Table 2(c) ; the f’s and k’s are related as follows: le, = f, + 4f, + f,‘, k, = -@CL + (3/WL’l, k,, = -(3/16)raf,,‘, where r is the P-P distance. As SLATER[3] has pointed out, the force field of equation (3) is less appropriate than that of equation (2) in the present case, since f,.and not Ic, is the effective valence stretching force constant for both potentials. For there to be a restoring force for angular deformation, k, must be positive, and hence f, must be negatvie, as is observed. From the frequencies of Table 2 the mean amplitudes of vibration can be obtained by straightforward methods; these are given in Table 2(d). They differ by a few percent from values which have been reported based on the observed frequencies [ 151 and on Pistorius’ frequencies [ 161. The constants of Table 2 are, or course, strictly valid only for CSz solutions of phosphorus. However, they are probably nearly correct also for phosphorus vapor, since the differences in band positions for solutions, liquid, and vapor are only slight 151. [14] S. BHWAVANTAM and T. VENKATARAYUDU,Proc. Indian Ad. [16] J. BAKKEN, Aclu Chm. Scat&. I.!& 694 (1968). [16] S. J. CYVIN, Acta Chm. Scar&. 18,1397 (1969).

Sci. AS, 119 (1938).