- Email: [email protected]
High-resolution Fourier transform infrared spectroscopy and analysis of the ν3 fundamental band of P4
14 May 1999
Chemical Physics Letters 305 Ž1999. 21–27
High-resolution Fourier transform infrared spectroscopy and analysis of the n 3 fundamental band of P4 V. Boudon a
a,)
b , E.B. Mkadmi b, H. Burger , G. Pierre ¨
a
Laboratoire de Physique de l’UniÕersite´ de Bourgogne–CNRS, B.P. 47 870, F-21078 Dijon, France b Anorganische Chemie, UniÕersitat-Gesamthochschule, D-42097 Wuppertal, Germany ¨ Received 25 January 1999; in final form 1 March 1999 In Memoriam Marco Haeser
Abstract We present the first high-resolution infrared absorption study of the n 3 fundamental of white phosphorus, P4 . This spectrum has been analyzed using the STDS ŽSpherical Top Data System. software. The band center lies at 466.286 cmy1. With the approximation Ž Bz . 3 s yB0r2, we found that the ground-state bond length is r 0 s 219.58 pm. This value is consistent with that of ab initio studies reported previously but significantly different from a value obtained from a Raman study. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In spite of its fundamental importance as a wellknown metastable modification of a ubiquitous, common element of technical importance the structure and spectroscopic parameters of the free P4 molecule are not known at high accuracy. Experimental studies related to the structure of P4 are Ži. an old gas phase electron diffraction study yielding r g f r 0 ŽP– ˚ w1x and Žii. a low-temperature X-ray P. s 2.21Ž2. A diffraction study of crystalline white phosphorus giv˚ w2x. Since ing a P–P bond length of 2.209 " 0.005 A P4 has no permanent dipole moment and is therefore not amenable to microwave spectroscopy, Žiii. the
) Corresponding author. Fax: q33 3 80 39 59 17; e-mail: [email protected]
vibration–rotation Raman spectrum of the n 2 band studied by Brassington et al. w3x provided the hitherto unique experimental information on the ground-state ŽGS. rotational constant B0 s 0.11015 " 0.00005 ˚ was calcmy1 , from which r 0 s 2.2228 " 0.0005 A culated. This apparently very accurate P–P bond length has been seriously called into question by two recent high-level ab initio calculations w4,5x. These defi˚ nitely rule out P–P bond lengths greater than 2.20 A, and their best estimates, after applying empirical ˚ w4x and re s 2.186 " corrections, are re s 2.194 A ˚ w5x. With the help of the ab initio anharmonic 0.001 A ˚ was force field a ‘true’ r 0 value, 2.191 " 0.001 A, ˚ shorter than predicted w5x, which is more than 0.03 A that deduced from the Raman experiment w3x. Less disputable but nevertheless numerically not consistent is the location of the three vibrational fundamentals of P4 . In the most recent Raman study
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 3 4 8 - 6
22
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
of P4 at 540 K w6x the following band centers have been determined:
n 1ŽA 1 . n 2 ŽE. n 3 ŽF2 .
600.51 " 0.01 cmy1 360.813 " 0.014 cmy1 466.925 " 0.009 cmy1
The wavenumbers of n 1 and n 3 represent the centers of the Q branches. The n 1 to n 3 wavenumbers differ by 5–6 cmy1 from those obtained at 580 K that were reported by Bosworth et al. w7x. The only infrared ŽIR.-active fundamental n 3 has been studied at low resolution in the gas phase at 503 K, and the band center was located at 464.5 cmy1 w8x. The authors of Ref. w5x do not endorse these figures and they believe that hardly any discrepancy exceeding a few cmy1 is possible with regard to their ‘best’ values, which are 625, 374 and 472 cmy1 , respectively, for n 1 , n 2 and n 3 . Part of this apparent discrepancy is ascribed to the fact that at the elevated temperatures of the relevant measurements a shift to small wavenumber is expected due both to vibrational and rotational population of highly excited states w5x. In order to measure a state-resolved rovibrational band of P4 in the IR, to determine the ground- and excited-state rotational constants B0 and B3 , to settle
the disagreement concerning the structure of P4 , and to determine the band center of n 3 , we have performed a high-resolution IR study of this band at the lowest temperature feasible, 413 K. We have resolved the Q branch and the tetrahedral fine structure of the J clusters in the P and R branches and unambiguously determined several spectroscopic parameters up to third order that enable a satisfactory reproduction of the experimental spectrum by a simulation. Here we report our results.
2. Experiment Measurements were performed using a Bruker 120HR interferometer employing an external stainless steel cell of 140 cm length and 8 cm inner diameter equipped with KBr windows. The parallel external beam of the interferometer was used in conjunction with an external detector chamber similar to the arrangement described in Ref. w9x. During the experiment the cell was heated to 1408C. The interferometer was equipped with a globar source, a 3.5 mm Mylar beam splitter and a liquid heliumcooled B:Si detector. A 12.5 mm low-pass filter was used, and the 370–800 cmy1 region studied. The
Fig. 1. Global view of the n 3 band of P4 .
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
23
Table 1 Effective parameters for the n 3 band of P4 . The standard deviation is given in parentheses, in the units of the last two digits X
Level
GS
n3 s1
Parameter
tVs4Ž sKX 4, n G . G v G v
V Ž K,n G .
s4Gv
sX 4G X y v
000A 1 000A 1
0
2Ž0,0A 1 . 4Ž0,0A 1 . 4Ž4,0A 1 . 0Ž0,0A 1 .
1
Valuercmy1
Notation of Robiette et al.
000A 1 000A 1
0.1128757Ž25. y2.0347 = 10y8
000A 1 001F2
000A 1 001F2
y4.5675 = 10y10 466.17262Ž20.
B0 yD 0 yŽ'15 r4'2 . D 0 t n3
1Ž1,0F1 .
001F2
001F2
2
2Ž0,0A 1 . 2Ž2,0E. 2Ž2,0F2 .
001F2 001F2 001F2
001F2 001F2 001F2
3
3Ž1,0F1 . 3Ž3,0F1 .
001F2
001F2
Fixed to 3'2 B0 z 3 with z 3 s y1r2 2.387Ž15. = 10y5 y4.893Ž36. = 10y5 2.080 = 10y4 y5.385Ž35. = 10y7
001F2
001F2
y3.342Ž26. = 10y7
656 lines fitted
Jma x s 63
s s 2.54 = 10y3 cmy1
Order
0 2
resolution Ž1rmaximum optical path difference. was adjusted to 6 = 10y3 cmy1 , and trapezoidal apodization applied. A total of 22 scans were collected, and a signal-to-noise ratio of the power spectrum of ca. 50 was achieved. Calibration was done with H 2 O lines taken from Ref. w10x in the 500–550 cmy1 region. Wavenumber accuracy is 1 = 10y3 cmy1 .
B3 y B0 yŽ1r2. a 220 y 6 a 224 yŽ3r4. a 220 q 6 a 224
Constraint
Fixed Fixed
Fixed
yŽ3'3 r4'2 .F110 Ž3'5 r2.F134
P4 was evaporated from a heated glass container in a flow of Ar passing beforehand through a capillary of 100 cm length and 0.6 mm diameter to avoid back diffusion. This container the temperature of which was adjusted to ca. 1208C was connected to the absorption cell. The absorption cell was continuously pumped, removed P4 being condensed in a
Fig. 2. Detailed portion in the R-branch of the n 3 band of P4 compared to the simulation. The J values are indicated. Symmetry assignments are shown for the RŽ34. cluster.
24
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
liquid nitrogen trap to which a pressure gauge was connected. A capillary in the vacuum line served to adjust the pumping efficiency such that a total gas pressure of ca. 3000 Pa was maintained in the absorption cell.
3. Theory X 4 molecules like phosphorus possess three normal modes of vibration, say n 1 , n 2 and n 3 , of respective symmetry A 1 , E and F2 in the Td point group w11,6x. n 3 ŽF2 . is the only IR-active fundamental Žthe two others are Raman active.. The theoretical model described below to develop the Hamiltonian operator is based on the tensorial formalism and vibrational extrapolation methods used in Dijon. These methods have already been explained for example in Ref. w12x. We recall here the basic principles and their application to the case of an X 4 tetrahedral molecule. If we consider an X 4 molecule for which the vibrational levels are grouped in a series of polyads designed by Pk Ž k s 0, . . . ,n., P0 being the ground state ŽGS., the Hamiltonian operator can be put in
the following form Žafter performing some contact transformations.: H P k 4 H s H P 0 ' GS4 q H P14 q PPP qH H P ny 14 q H P n4 . q PPP qH
Ž 1.
Terms like H P k 4 contain rovibrational operators which have no matrix elements within the Pk X ) k basis sets. The effective Hamiltonian for polyad Pn is obtained by projecting H in the Pn Hilbert subspace, i.e. ² Pn: q H²PP14n : H ² P n : s P ² P n :H P ² P n :s HGS4
q PPP qH²PPk 4n : q PPP qH²PPnyn :14 q H²PPn4n : .
Ž 2. The different terms are written in the form X
H P k 4 s
Ý
tVs4Ž sKX 4, n G . G v G v
all indexes X
= ´ V V sv4ŽGsX v4 G v . G m R V Ž K , n G .
Ž A 1.
.
Ž 3.
X equation, the tVs4Ž sKX 4, n G . GX v G v are v ŽX G v G v . G and determined. ´ V V s4 s 4
In this the parameters to be R V Ž K , n G . are vibrational and rotational operators of respective degree V v and V . Their construction is described in Ref. w12x. Again, the vibrational operators only have
Fig. 3. Detailed portion in the Q-branch of the n 3 band of P4 compared to the simulation. The sub-bandhead labels according to Fig. 2 of Ref. w15x are indicated.
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
matrix elements within the Pk X ( k basis sets. The order of each individual term is V q V v y 2. Such a Hamiltonian development scheme enables the treatment of any polyad system. In this work however, since we only consider an isolated fundamental band, we will use only the two following effective Hamiltonians: Ø The GS effective Hamiltonian, ²GS: . H ²GS: s HGS4
Ž 4.
Ø The n 3 stretching fundamental effective Hamiltonian, ²n 3: H ² n 3 : s HGS4 q H²nn343 : .
Ž 5.
A dipole moment operator is developed in the same way, but, in the case we study in this Letter, it reduces to a single trivial operator. As we do not consider absolute intensities, the corresponding parameter is set to one. We use a vibrational basis in the coupled form
CvŽC v . :s CÕŽ1A 1 . m CÕŽ2l 2 ,C 2 . m CÕŽ3l 3 , n 3 C 3 .
25
parameters from Ref. w5x and with the band center placed at the maximum of the most prominent Q branch feature of Fig. 1. As X 4 molecules only have one F2 vibration, the zeta sum rule w14x reduces to 1Ž1,0 F1 .F 2 F 2 z 3 s y1r2. We thus approximated t3434 s 3'2 Ž Bz . 3 by y3B0r '2 . Several trials were made 2Ž2,0 E .F 2 F 2 2Ž2,0 F 2 .F 2 F 2 for different values of t3434 , t3434 being set to zero. This procedure enabled us to assign many lines in the P and R branches. Since the Q branch consists of several hundreds of unresolved transitions in a very small frequency range, this was too dense to allow any assignment in this region. Then, it was possible to perform a parameter fit using a least-squares fit method. The fit was realized using the following constraints: 1Ž1,0 F1 .F 2 F 2 Ø t3434 was fixed to y3B0r '2 and B0 was fitted. In view of the smallness of the difference
ŽC v .
;, Ž 6.
the C s being harmonic oscillator wavefunctions. The Hamiltonian and dipole moment matrix elements are calculated in the coupled rovibrational basis
CvŽC v . m Cr Ž J , nC .
ŽG .
;,
Ž 7.
Cr Ž J, nC . being a rotational wavefunction. 4. Analysis of the n 3 spectrum All the calculations and fits were realized using the STDS ŽSpherical Top Data System. program chain developed in Dijon Žsee Ref. w13x.. As this software is designed for the study of tetrahedral XY4 molecules with four normal modes of vibration Ž n 1ŽA 1 ., n 2 ŽE., n 3 ŽF2 . and n4ŽF2 .. we simply used it by considering the n 3 ŽF2 . fundamental of P4 as a n 3 ŽF2 . fundamental of an XY4 molecule. In order to get a correct vibrational partition function, the n4 frequency was set to an arbitrarily high value. Here, HGS4 is developed to second order and Hn 34 is developed to third order. The assignments of the n 3 band of P4 were realized as follows. We first made preliminary simulations using ab initio GS
Fig. 4. Reduced energies for the n 3 band of P4 . The top diagram shows only the levels corresponding to lines that could be observed and assigned. The values of the R quantum number are indicated Ž R s J I l where l is the vibrational quantum number..
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
26
Ž B3 y B0 ., 2.39 = 10y5 cmy1 , it does not matter for the conclusions whether B0 or B3 is used. Ø the other GS parameters where fixed to their ab initio values, 2Ž2,0 F 2 .F 2 F 2 Ø t3434 was fixed to zero in a first step since it is not possible to fit it simultaneously with 2Ž2,0 E .F 2 F 2 t3434 if no Q branch assignments are available. Since the Qbranch shows a characteristic sub-bandhead structure ŽFig. 3. w15x, it was possible to adjust 2Ž2,0 F 2 .F 2 F 2 manually the value of t3434 as follows: this parameter was set to different values and each time the other parameters were fitted again with the same constraints as above. This was repeated until the Q branch simulation was correct. Table 1 shows the result of the final fit. Six hundred and fifty six P and R transitions were assigned and the root-meansquare deviation is 2.54 = 10y3 cmy1 . The maximum J values that could be assigned in the P and R branches were 54 and 63, respectively. As can be seen in Fig. 2 and Fig. 3, the simulation is very satisfactory. Fig. 4 shows the reduced energy levels obtained by subtracting the scalar terms, i.e. V Ž0 ,0 A 1 .A 1 A 1 Ered s E y Ý t0404 Ž J Ž J q 1. .
Vr2
V 2
s E y B0 J Ž J q 1 . q D 0 J 2 Ž J q 1 . .
5. Discussion The band center m can be calculated through w16x:
'2 3
1Ž1 ,0 F1 .F 2 F 2 t3434
E . F2 F2 2Ž2 ,0 F 2.F 2 F 2 q 25 Ž t23Ž42,0 q t3434 . q PPP , 34
Ž 9.
and we find m s 466.286 cmy1 .
r0 s
(
h 2
8p cm P B0
s 219.58 pm ,
Ž 11 .
Ž m P s 5.143397 = 10y2 6 kg. which is close to the 219.1 pm ab initio value w5x. Moreover, if we make two new fits by varying z 3 by "1% Žwhich would be a pessimistic error; in CH 4 , for instance, the zeta sum rule is verified with a much better precision w17x., we obtain r 0 s 219.22 and 219.95 pm, respectively. These results seem to definitely rule out the high r 0 value derived from the Raman study of Ref. w3x. The hot bands that appear in the experimental spectrum could not be analyzed here because of their complexity. They should be dominated by n 3 q n 2 y n 2 since n 2 ŽE. is the lowest fundamental band at 360.813 cmy1 w6,3x. The understanding of this hot band would probably require first the analysis of the n 2 q n 3 combination band.
Ž 8.
The top of the figure displays only levels corresponding to the transitions that could be assigned.
0Ž0 ,0 A 1 .F 2 F 2 m s t3434 y
assumption is based on two approximations, namely B0 , B3 , which is correct within 0.02% Žsee Table 1., and the neglect of any Õ-dependance of z 3 , this appears to be justified. If we calculate the GS bond length using our B0 value, we get
6. Conclusions We have been able to locate the n 3 fundamental band of P4 unambiguously in the gas phase, and resolved the rotational structure. We conclude that, with the assumption that Ž Bz . 3 , yB0r2, we can support the ab initio structural parameters for P4 w4,5x rather than those from Raman spectroscopy w3x. The difference between our n 3 value and the ab initio calculation w5x is 6 cmy1 which is an acceptable discrepancy. This corresponds to ‘a few cmy1 ’ as mentioned in Ref. w5x.
Ž 10 .
The strict selection rules that apply to spherical top rovibrational spectra make it impossible to determine B0 from GS combination differences Žthis is the same as for the axial rotational constant in symmetric top molecules.. Thus, as mentioned above, we were forced to make the assumption that Ž Bz . 3 , B0 z 3 which is only approximative. Although this
Acknowledgements The ‘Region Bourgogne’ is gratefully acknowl´ edged for its support to the ‘Laboratoire de Physique de l’Universite´ de Bourgogne’. EBM gratefully acknowledges support by the Max-Planck-Gesellschaft zur Forderung der Wissenschaften. ¨
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
References w1x L.B. Maxwell, S.B. Hendricks, V. Mosley, J. Chem. Phys. 3 Ž1935. 699. w2x A. Simon, H. Borrmann, H. Craubner, Phosphorus Sulfur 30 Ž1987. 507. w3x N. Brassington, H. Edwards, D. Long, J. Raman. Spectrosc. 11 Ž1981. 346. w4x M. Haser, O. Treutler, J. Chem. Phys. 102 Ž1995. 3703. ¨ w5x B. Persson, R. Taylor, T. Lee, J. Chem. Phys. 107 Ž1997. 5051. w6x H. Edwards, J. Mol. Struct. 295 Ž1993. 95. w7x Y.M. Bosworth, R.J.H. Clark, D.M. Rippon, J. Mol. Spectrosc. 46 Ž1970. 240. w8x H. Gutowsky, C. Hoffman, J. Am. Chem. Soc. 72 Ž1950. 5751.
27
w9x M. Birk, M. Winnewisser, E.A. Cohen, J. Mol. Spectrosc. 136 Ž1985. 402. w10x Guelachvili, K.N. Rao, Handbook of Infrared Standards, Academic Press, San Diego, CA, 1986. w11x R. McDowell, Spectrochim. Acta 27A Ž1971. 773. w12x J.-P. Champion, M. Loete, ¨ G. Pierre, in: K.N. Rao, A. Weber ŽEds.., Spectroscopy of the Earth’s Atmosphere and Interstellar Medium, Academic Press, San Diego, CA, 1992, pp. 339–422. w13x C. Wenger, J.-P. Champion, J. Quantum Spectrosc. Radiat. Transfer 59 Ž1998. 471. w14x D. Lepard, Can. J. Phys. 44 Ž1966. 461. w15x E. Brock, B. Krohn, R. McDowell, C. Patterson, D. Smith, J. Mol. Spectrosc. 76 Ž1979. 301. w16x B. Bobin, K. Fox, J. Phys. ŽParis. 34 Ž1973. 571. w17x D.L. Gray, A.G. Robiette, Mol. Phys. 37 Ž1979. 1901.
Chemical Physics Letters 305 Ž1999. 21–27
High-resolution Fourier transform infrared spectroscopy and analysis of the n 3 fundamental band of P4 V. Boudon a
a,)
b , E.B. Mkadmi b, H. Burger , G. Pierre ¨
a
Laboratoire de Physique de l’UniÕersite´ de Bourgogne–CNRS, B.P. 47 870, F-21078 Dijon, France b Anorganische Chemie, UniÕersitat-Gesamthochschule, D-42097 Wuppertal, Germany ¨ Received 25 January 1999; in final form 1 March 1999 In Memoriam Marco Haeser
Abstract We present the first high-resolution infrared absorption study of the n 3 fundamental of white phosphorus, P4 . This spectrum has been analyzed using the STDS ŽSpherical Top Data System. software. The band center lies at 466.286 cmy1. With the approximation Ž Bz . 3 s yB0r2, we found that the ground-state bond length is r 0 s 219.58 pm. This value is consistent with that of ab initio studies reported previously but significantly different from a value obtained from a Raman study. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In spite of its fundamental importance as a wellknown metastable modification of a ubiquitous, common element of technical importance the structure and spectroscopic parameters of the free P4 molecule are not known at high accuracy. Experimental studies related to the structure of P4 are Ži. an old gas phase electron diffraction study yielding r g f r 0 ŽP– ˚ w1x and Žii. a low-temperature X-ray P. s 2.21Ž2. A diffraction study of crystalline white phosphorus giv˚ w2x. Since ing a P–P bond length of 2.209 " 0.005 A P4 has no permanent dipole moment and is therefore not amenable to microwave spectroscopy, Žiii. the
) Corresponding author. Fax: q33 3 80 39 59 17; e-mail: [email protected]
vibration–rotation Raman spectrum of the n 2 band studied by Brassington et al. w3x provided the hitherto unique experimental information on the ground-state ŽGS. rotational constant B0 s 0.11015 " 0.00005 ˚ was calcmy1 , from which r 0 s 2.2228 " 0.0005 A culated. This apparently very accurate P–P bond length has been seriously called into question by two recent high-level ab initio calculations w4,5x. These defi˚ nitely rule out P–P bond lengths greater than 2.20 A, and their best estimates, after applying empirical ˚ w4x and re s 2.186 " corrections, are re s 2.194 A ˚ w5x. With the help of the ab initio anharmonic 0.001 A ˚ was force field a ‘true’ r 0 value, 2.191 " 0.001 A, ˚ shorter than predicted w5x, which is more than 0.03 A that deduced from the Raman experiment w3x. Less disputable but nevertheless numerically not consistent is the location of the three vibrational fundamentals of P4 . In the most recent Raman study
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 3 4 8 - 6
22
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
of P4 at 540 K w6x the following band centers have been determined:
n 1ŽA 1 . n 2 ŽE. n 3 ŽF2 .
600.51 " 0.01 cmy1 360.813 " 0.014 cmy1 466.925 " 0.009 cmy1
The wavenumbers of n 1 and n 3 represent the centers of the Q branches. The n 1 to n 3 wavenumbers differ by 5–6 cmy1 from those obtained at 580 K that were reported by Bosworth et al. w7x. The only infrared ŽIR.-active fundamental n 3 has been studied at low resolution in the gas phase at 503 K, and the band center was located at 464.5 cmy1 w8x. The authors of Ref. w5x do not endorse these figures and they believe that hardly any discrepancy exceeding a few cmy1 is possible with regard to their ‘best’ values, which are 625, 374 and 472 cmy1 , respectively, for n 1 , n 2 and n 3 . Part of this apparent discrepancy is ascribed to the fact that at the elevated temperatures of the relevant measurements a shift to small wavenumber is expected due both to vibrational and rotational population of highly excited states w5x. In order to measure a state-resolved rovibrational band of P4 in the IR, to determine the ground- and excited-state rotational constants B0 and B3 , to settle
the disagreement concerning the structure of P4 , and to determine the band center of n 3 , we have performed a high-resolution IR study of this band at the lowest temperature feasible, 413 K. We have resolved the Q branch and the tetrahedral fine structure of the J clusters in the P and R branches and unambiguously determined several spectroscopic parameters up to third order that enable a satisfactory reproduction of the experimental spectrum by a simulation. Here we report our results.
2. Experiment Measurements were performed using a Bruker 120HR interferometer employing an external stainless steel cell of 140 cm length and 8 cm inner diameter equipped with KBr windows. The parallel external beam of the interferometer was used in conjunction with an external detector chamber similar to the arrangement described in Ref. w9x. During the experiment the cell was heated to 1408C. The interferometer was equipped with a globar source, a 3.5 mm Mylar beam splitter and a liquid heliumcooled B:Si detector. A 12.5 mm low-pass filter was used, and the 370–800 cmy1 region studied. The
Fig. 1. Global view of the n 3 band of P4 .
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
23
Table 1 Effective parameters for the n 3 band of P4 . The standard deviation is given in parentheses, in the units of the last two digits X
Level
GS
n3 s1
Parameter
tVs4Ž sKX 4, n G . G v G v
V Ž K,n G .
s4Gv
sX 4G X y v
000A 1 000A 1
0
2Ž0,0A 1 . 4Ž0,0A 1 . 4Ž4,0A 1 . 0Ž0,0A 1 .
1
Valuercmy1
Notation of Robiette et al.
000A 1 000A 1
0.1128757Ž25. y2.0347 = 10y8
000A 1 001F2
000A 1 001F2
y4.5675 = 10y10 466.17262Ž20.
B0 yD 0 yŽ'15 r4'2 . D 0 t n3
1Ž1,0F1 .
001F2
001F2
2
2Ž0,0A 1 . 2Ž2,0E. 2Ž2,0F2 .
001F2 001F2 001F2
001F2 001F2 001F2
3
3Ž1,0F1 . 3Ž3,0F1 .
001F2
001F2
Fixed to 3'2 B0 z 3 with z 3 s y1r2 2.387Ž15. = 10y5 y4.893Ž36. = 10y5 2.080 = 10y4 y5.385Ž35. = 10y7
001F2
001F2
y3.342Ž26. = 10y7
656 lines fitted
Jma x s 63
s s 2.54 = 10y3 cmy1
Order
0 2
resolution Ž1rmaximum optical path difference. was adjusted to 6 = 10y3 cmy1 , and trapezoidal apodization applied. A total of 22 scans were collected, and a signal-to-noise ratio of the power spectrum of ca. 50 was achieved. Calibration was done with H 2 O lines taken from Ref. w10x in the 500–550 cmy1 region. Wavenumber accuracy is 1 = 10y3 cmy1 .
B3 y B0 yŽ1r2. a 220 y 6 a 224 yŽ3r4. a 220 q 6 a 224
Constraint
Fixed Fixed
Fixed
yŽ3'3 r4'2 .F110 Ž3'5 r2.F134
P4 was evaporated from a heated glass container in a flow of Ar passing beforehand through a capillary of 100 cm length and 0.6 mm diameter to avoid back diffusion. This container the temperature of which was adjusted to ca. 1208C was connected to the absorption cell. The absorption cell was continuously pumped, removed P4 being condensed in a
Fig. 2. Detailed portion in the R-branch of the n 3 band of P4 compared to the simulation. The J values are indicated. Symmetry assignments are shown for the RŽ34. cluster.
24
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
liquid nitrogen trap to which a pressure gauge was connected. A capillary in the vacuum line served to adjust the pumping efficiency such that a total gas pressure of ca. 3000 Pa was maintained in the absorption cell.
3. Theory X 4 molecules like phosphorus possess three normal modes of vibration, say n 1 , n 2 and n 3 , of respective symmetry A 1 , E and F2 in the Td point group w11,6x. n 3 ŽF2 . is the only IR-active fundamental Žthe two others are Raman active.. The theoretical model described below to develop the Hamiltonian operator is based on the tensorial formalism and vibrational extrapolation methods used in Dijon. These methods have already been explained for example in Ref. w12x. We recall here the basic principles and their application to the case of an X 4 tetrahedral molecule. If we consider an X 4 molecule for which the vibrational levels are grouped in a series of polyads designed by Pk Ž k s 0, . . . ,n., P0 being the ground state ŽGS., the Hamiltonian operator can be put in
the following form Žafter performing some contact transformations.: H P k 4 H s H P 0 ' GS4 q H P14 q PPP qH H P ny 14 q H P n4 . q PPP qH
Ž 1.
Terms like H P k 4 contain rovibrational operators which have no matrix elements within the Pk X ) k basis sets. The effective Hamiltonian for polyad Pn is obtained by projecting H in the Pn Hilbert subspace, i.e. ² Pn: q H²PP14n : H ² P n : s P ² P n :H P ² P n :s HGS4
q PPP qH²PPk 4n : q PPP qH²PPnyn :14 q H²PPn4n : .
Ž 2. The different terms are written in the form X
H P k 4 s
Ý
tVs4Ž sKX 4, n G . G v G v
all indexes X
= ´ V V sv4ŽGsX v4 G v . G m R V Ž K , n G .
Ž A 1.
.
Ž 3.
X equation, the tVs4Ž sKX 4, n G . GX v G v are v ŽX G v G v . G and determined. ´ V V s4 s 4
In this the parameters to be R V Ž K , n G . are vibrational and rotational operators of respective degree V v and V . Their construction is described in Ref. w12x. Again, the vibrational operators only have
Fig. 3. Detailed portion in the Q-branch of the n 3 band of P4 compared to the simulation. The sub-bandhead labels according to Fig. 2 of Ref. w15x are indicated.
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
matrix elements within the Pk X ( k basis sets. The order of each individual term is V q V v y 2. Such a Hamiltonian development scheme enables the treatment of any polyad system. In this work however, since we only consider an isolated fundamental band, we will use only the two following effective Hamiltonians: Ø The GS effective Hamiltonian, ²GS: . H ²GS: s HGS4
Ž 4.
Ø The n 3 stretching fundamental effective Hamiltonian, ²n 3: H ² n 3 : s HGS4 q H²nn343 : .
Ž 5.
A dipole moment operator is developed in the same way, but, in the case we study in this Letter, it reduces to a single trivial operator. As we do not consider absolute intensities, the corresponding parameter is set to one. We use a vibrational basis in the coupled form
CvŽC v . :s CÕŽ1A 1 . m CÕŽ2l 2 ,C 2 . m CÕŽ3l 3 , n 3 C 3 .
25
parameters from Ref. w5x and with the band center placed at the maximum of the most prominent Q branch feature of Fig. 1. As X 4 molecules only have one F2 vibration, the zeta sum rule w14x reduces to 1Ž1,0 F1 .F 2 F 2 z 3 s y1r2. We thus approximated t3434 s 3'2 Ž Bz . 3 by y3B0r '2 . Several trials were made 2Ž2,0 E .F 2 F 2 2Ž2,0 F 2 .F 2 F 2 for different values of t3434 , t3434 being set to zero. This procedure enabled us to assign many lines in the P and R branches. Since the Q branch consists of several hundreds of unresolved transitions in a very small frequency range, this was too dense to allow any assignment in this region. Then, it was possible to perform a parameter fit using a least-squares fit method. The fit was realized using the following constraints: 1Ž1,0 F1 .F 2 F 2 Ø t3434 was fixed to y3B0r '2 and B0 was fitted. In view of the smallness of the difference
ŽC v .
;, Ž 6.
the C s being harmonic oscillator wavefunctions. The Hamiltonian and dipole moment matrix elements are calculated in the coupled rovibrational basis
CvŽC v . m Cr Ž J , nC .
ŽG .
;,
Ž 7.
Cr Ž J, nC . being a rotational wavefunction. 4. Analysis of the n 3 spectrum All the calculations and fits were realized using the STDS ŽSpherical Top Data System. program chain developed in Dijon Žsee Ref. w13x.. As this software is designed for the study of tetrahedral XY4 molecules with four normal modes of vibration Ž n 1ŽA 1 ., n 2 ŽE., n 3 ŽF2 . and n4ŽF2 .. we simply used it by considering the n 3 ŽF2 . fundamental of P4 as a n 3 ŽF2 . fundamental of an XY4 molecule. In order to get a correct vibrational partition function, the n4 frequency was set to an arbitrarily high value. Here, HGS4 is developed to second order and Hn 34 is developed to third order. The assignments of the n 3 band of P4 were realized as follows. We first made preliminary simulations using ab initio GS
Fig. 4. Reduced energies for the n 3 band of P4 . The top diagram shows only the levels corresponding to lines that could be observed and assigned. The values of the R quantum number are indicated Ž R s J I l where l is the vibrational quantum number..
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
26
Ž B3 y B0 ., 2.39 = 10y5 cmy1 , it does not matter for the conclusions whether B0 or B3 is used. Ø the other GS parameters where fixed to their ab initio values, 2Ž2,0 F 2 .F 2 F 2 Ø t3434 was fixed to zero in a first step since it is not possible to fit it simultaneously with 2Ž2,0 E .F 2 F 2 t3434 if no Q branch assignments are available. Since the Qbranch shows a characteristic sub-bandhead structure ŽFig. 3. w15x, it was possible to adjust 2Ž2,0 F 2 .F 2 F 2 manually the value of t3434 as follows: this parameter was set to different values and each time the other parameters were fitted again with the same constraints as above. This was repeated until the Q branch simulation was correct. Table 1 shows the result of the final fit. Six hundred and fifty six P and R transitions were assigned and the root-meansquare deviation is 2.54 = 10y3 cmy1 . The maximum J values that could be assigned in the P and R branches were 54 and 63, respectively. As can be seen in Fig. 2 and Fig. 3, the simulation is very satisfactory. Fig. 4 shows the reduced energy levels obtained by subtracting the scalar terms, i.e. V Ž0 ,0 A 1 .A 1 A 1 Ered s E y Ý t0404 Ž J Ž J q 1. .
Vr2
V 2
s E y B0 J Ž J q 1 . q D 0 J 2 Ž J q 1 . .
5. Discussion The band center m can be calculated through w16x:
'2 3
1Ž1 ,0 F1 .F 2 F 2 t3434
E . F2 F2 2Ž2 ,0 F 2.F 2 F 2 q 25 Ž t23Ž42,0 q t3434 . q PPP , 34
Ž 9.
and we find m s 466.286 cmy1 .
r0 s
(
h 2
8p cm P B0
s 219.58 pm ,
Ž 11 .
Ž m P s 5.143397 = 10y2 6 kg. which is close to the 219.1 pm ab initio value w5x. Moreover, if we make two new fits by varying z 3 by "1% Žwhich would be a pessimistic error; in CH 4 , for instance, the zeta sum rule is verified with a much better precision w17x., we obtain r 0 s 219.22 and 219.95 pm, respectively. These results seem to definitely rule out the high r 0 value derived from the Raman study of Ref. w3x. The hot bands that appear in the experimental spectrum could not be analyzed here because of their complexity. They should be dominated by n 3 q n 2 y n 2 since n 2 ŽE. is the lowest fundamental band at 360.813 cmy1 w6,3x. The understanding of this hot band would probably require first the analysis of the n 2 q n 3 combination band.
Ž 8.
The top of the figure displays only levels corresponding to the transitions that could be assigned.
0Ž0 ,0 A 1 .F 2 F 2 m s t3434 y
assumption is based on two approximations, namely B0 , B3 , which is correct within 0.02% Žsee Table 1., and the neglect of any Õ-dependance of z 3 , this appears to be justified. If we calculate the GS bond length using our B0 value, we get
6. Conclusions We have been able to locate the n 3 fundamental band of P4 unambiguously in the gas phase, and resolved the rotational structure. We conclude that, with the assumption that Ž Bz . 3 , yB0r2, we can support the ab initio structural parameters for P4 w4,5x rather than those from Raman spectroscopy w3x. The difference between our n 3 value and the ab initio calculation w5x is 6 cmy1 which is an acceptable discrepancy. This corresponds to ‘a few cmy1 ’ as mentioned in Ref. w5x.
Ž 10 .
The strict selection rules that apply to spherical top rovibrational spectra make it impossible to determine B0 from GS combination differences Žthis is the same as for the axial rotational constant in symmetric top molecules.. Thus, as mentioned above, we were forced to make the assumption that Ž Bz . 3 , B0 z 3 which is only approximative. Although this
Acknowledgements The ‘Region Bourgogne’ is gratefully acknowl´ edged for its support to the ‘Laboratoire de Physique de l’Universite´ de Bourgogne’. EBM gratefully acknowledges support by the Max-Planck-Gesellschaft zur Forderung der Wissenschaften. ¨
V. Boudon et al.r Chemical Physics Letters 305 (1999) 21–27
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