Ground state reduced potential curves (RPC) of BeF, CS, SiN, P2, SiS and GeO

SpectrochimicaActa, Vol.24.~.,pp. 259 to 264. PergamonPress 1968. Printed in NorthernIreland

6round state reduced potential curves (RPC) of BeF, (IS, SiN, P2, SiS and Ge0 F. JEN5 I n s t i t u t e of Physical Chemistry, Czechoslovak Academy of Sciences, Prague

(Received 29 M a y 1967) Abstract--Ground state reduced potential curves calculated from the R y d b e r g - K l e i n - R e e s Vandorslieo potential curves are examined for the molecules of BeF, CS, SiN, P2, SiS, and GeO a n d found to satisfy the general rule discovered in previous study of reduced potential curves of diatomics. INTRODUCTI01~

THE concept of the reduced potential curve (RPC) was recently introduced by the author [1] in order to obtain a unique scheme for a comparative study of diatomic potentials. I n previous papers [1-5] (for bibliography cf. Ref. [5]), a large n u m b e r of diatomic molecules were studied. The regularities observed for the ground state RPC's seem to represent a general law which, in principle, could be deduced from the mathematical properties of the electronic Schroedinger operators. A verification of the general validity of this law for as m a n y molecules as possible naturally seems to be of interest. The main difficulty here is the following: (1) reliable experimental data is in fact known only for a quite limited number of molecules; (2) even ff the necessary data is available, the RPC's can, for m a n y molecules, be constructed only in a relatively small neighbourhood of the equilibrium distance since the spectrum is not usually measured up to sufficiently high values of the vibrational q u a n t u m number, v. As new spectroscopic data appears in the literature, it becomes possible to s t u d y the RPC's of new series of molecules. I n the present paper, the RPC's of the molecules of BeF, CS, SiN, P2, SiS, and GeO are examined. [1] [2] [3] [4] [5]

F. F. F. F. F.

JEN(~ and J. PZtVA, Coll. Czech. Chem. Commun. 28, 1449 (1963). J~.N~, Coll. Czech. Chem. Commun. 29, 1521 (1964). J~.N~, Coll. Czech. Chem. Commun. 30, 3772 (1965). JEN~, J . Mol. Spectry 24, 284 (1967). JEN~, J . Chem. Phys. 47, 127 (1967). 259

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REDUCED POTENTIAL CURVES Reduced potential energy, u, and reduced internuclear distance, p, respectively, are defined by the following equations: (1)

u = U/D e r - - [ 1 - - exp ( - r/p,j)]p,~ P = r e - [1 - exp (-r/p,~)]p~j

P~' ---- 1 - - e x p ( - - r J p , ¢ )

r , - - \ ke /

(2)

j

(3)

Here K ~ 3.96 and U and r are the potential energy and the internuclear distance, respectively, r, is the equilibrium internuclear distance and D e denotes the depth of the potential curve at r e. k6 is the force constant, which is determined from the equation k e - - (d2V/dr2)~=r, -= I~O~,~

(4)

/~ and we being the reduced mass and the "vibrational harmonic frequency", respectively. The validity of equation (4) is generally assumed in molecular spectroscopy and m a y be accepted as a reliable approximation, at least for the ground states. The RPC's are constructed from the Rydberg-Klein-Rees-Vanderslice (RKRV) potential curves [6] calculated from the measured spectral lines; the reliability of the R K R V potential curves, in absence of nonadiabatic perturbations, is supported b y the recent accurate calculations of KoLos and WOLNIEWtCZ [7]. I n the present paper, we concentrate on the ground states. Reliable R K R V curves have [so far] been calculated for the following 22 molecules: H2, LiH, B e l l +, OH, H F , C2, N2, 02, CN, CO, NO, IC1, IBr, I2, SiC, SO, $2, Au2, I n H , Gall, T1H, AgH. For these molecules, the molecular constants, re, De, and ke are known to sufficient accuracy so t h a t the RPC's could be constructed. General regularities were observed which lead to the following RPC scheme: (1) W i t h growing values of atomic numbers, the values of the corresponding function 1 + u ( p ) increase in the left branch (i.e. for p < 1) and so increase the values of the absolute value of its first derivative, while, in the right branch (p > 1), the values of 1 + u ( p ) and its first derivative decrease in such a way, that, graphically, the ground state RPC "is broadened and t u r n e d to the right around the m i n i m u m " as the values of the atomic numbers increase. There is only one exception, namely the rare gases; the coinciding RPC's of the rare gas diatomic systems (obtained from Buckingham and Lennard-Jones potentials or from ab i n i t i o theoretical [6] J. T. VANDERSLICE,E. A. MASON,W. G. M~SCH and R. E. LIPPINCOTT,J. Mol. Spectry 3, 17 (1959); ST. WEISS.'~AN, J. T. VANDERSLICEand R. BAThetiC, J. Chem. Phys. 39, 2226 (1963). [7] W. Kor.os and L. WOLNIEWICZ.J. Chem. Phys. 43, 2429 (1965).

Ground state reduced potential curves (RPC}

261

calculations) form the right b o u n d a r y of the permissible R P C area; the left b o u n d a r y is formed b y the R P C of H 2 and the R P C ' s of all other dia¢omics lie between these two curves in the corresponding order. Thus, for r > r,, the reduced attractive force decreases with growing values of the atomic numbers (and is lowest for the rare gases) whereas, for r <2 r,, the reduced repulsive force slightly increases at the same time.

(2) The differences between the R P C ' s belonging to molecules possessing adjacent values of atomic numbers (further referred to as adjacent molecules) are v e r y small so t h a t the R P C ' s coincide approximately. The differences (in p) between the R P C ' s are very small in the repulsive branch. In the range studied, the ground state R P C ' s never intersect.

(3) The relative differences between the ground state R P C ' s belonging to molecules possessing different values of atomic numbers are considerably smaller in the region of high atomic numbers. Table 1. Values of molecular constants Molecule BeF. CS SiN P2 SiS GeO

Z1 4 6 7 15 14 8

Z~

r6 (.~)

De (eV)

ke (eV//~ ~)

(APt1)

9 16 14 15 16 32

1.3614 1-5349 1.5718 1.8939 1.9288 1-6507

5.378 7.880 4-571 5-081 6.640 6.960

36.0026 52.9870 45.5167 34.7275 30.8222 46.9814

0.68697 0.96324 1.38949 1.66992 1.30001 1-17042

A qualitative verification of the R P C scheme was obtained for about fifty other molecules, in a wide range of internuclear distance, in approximating the R K R V curve b y the HULBU~T--HIRsCHF~.LDV.R [8] potential function which had been found to yield the best fit to the R K R V curves [9]. As the R K R V curves for other molecules can be calculated, the validity of the R P C scheme can further be checked directly. In the present paper, the R P C ' s of BeF, CS, SiN, P2, SiS, and GeO are examined since, for these molecules, the R P C ' s could be calculated at least up to a b o u t (u ~- 1) ~ 0.25 and the experimental data seems sufficiently accurate. The R K R V curves were taken from [10] where the bibliography on the experimental data m a y also be found. The values of molecular constants are contained in Table 1. The value of the binding energy, D~, usually presents the main problem. The value of the binding energy of P~ had been uncertain for a long time; it seems now to be unequivocally fixed [11, 12] to 5.079 eV [8] H. M. HULBURT and J. O. ~.IRSCIiFI~LDER,J. Chem. Phys. 9, 61 (1941). [9] D. STEELE, E. R. LIPPI•COTT a n d J. T. VA-~DERSLICE, Rev. Mod. Phys. 84, 239 (1962). [10] R. B. SING~ and D. K. RAI, _rndian J. Pure Appl. Phys. 4, 102 (1966); K. P. R. I ~ R , R. B. SI~GH, and D. K. RxI, J. Chem. Phys. 48, 3570 (1965). [11] A. G. GAYDO~, Dissociation Energies and Spectra of Diatomic Molecules. Chapman & Hall (1953). [12] K. K. A. GINGERXCH,J. Chem. Phys. 44, 1716 (1966).

262

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and this value was used in the present paper. The value of 5.378 eV for De(BeF ) was accepted as recommended by HE~ZBERG [13] and GAYDON [11]. The value Do°(BeF) ---- 5.85 eV, advocated by I-IILDEBRA:ND and MURAD [14], implies a positive curvature in the Birge-Sponer curve as suggested by TXTEVSKI [15]; this anomaly, however, does not seem to be sufficiently confirmed by experiment. We note t h a t minor inaccuracies in the value of D e could not change the results significantly owing to the relative insensitivity of the RPC to the value of De[16]. ~:~ESU'LTS AND ])ISCUSSIO:~T

All the RPC's fit excellently in the RPC scheme (Figs. 1 and 2) in accord with the general rule discovered previously. The RPC of P~ practically coincides with the RPC of the adjacent molecule of SiS and also with the RPC of $2, reported previously [4]; the RPC of CS approximately coincides with the RPC of the adjacent molecule of SiN and with the RPC of SO reported in [4]. The agreement is surprisingly good; we note t h a t the RPC scheme is supposed [3] to hold strictly only for the exact adiabatic potentials. We see t h a t the value De(BeF ) ~-- 5-378 eV seems to be in order. The RPC method is not sensitive enough to decide whether there is some minor error in this

2.5

2.0

1.5 u+l

0~"

0.8

Fig. 1. Ground state reduced potential curves (repulsive branch). 1, It2; 2, P~; 3, Is; 4, rare gases; I, BeF; A, CS; O, SiN; O, SiS; ~, GeO. [13] G. HERZBER(~,Molecular Spectra and Molecular Structure, Part I. Spectra of Diatomic Molecules. Van Nostrand (1950). [14] D. L. H~D~.BR~D and E. MURAD, J. Chem. Phys. 44, 1524 (1966). [15] I. TATEVSKI,Opt. Spektroskoplja 5, 520 (1958). [16] F. JENC, Coll. Czech. Chem. Commun. 29, 1507 (1964).

Ground state reduced potential curves (RPC)

263

O~

0"2

u+l 14

1"5

15

I'Z

1, H a ; reduced potential curves (attractive branch). 5, I 2 ; 6, r a r e g a s e s ; A , C S ; e , S i N ; O , S i S ; + , G e O .

F i g . 2. G r o u n d s t a t e s 2, B o F ; 3, N2; 4, P g ;

T a b l e 2. R e d u c e d p o t e n t i a l c u r v e o f P ~ p

u-I- 1

p

0.648 0"652 0.655 0.659 0.663 0.668 0.672 0'676 0"681 0.687 0.692 0.697 0.702 0.707 0.714 0.720 0'726 0.734 0.742 0.749 0.758 0.767 0.776 0.787 0.798 0.812 0.827 0.843 0"865 0"891 0"935

0.514 0"499 0*484 0.469 0-454 0"439 0-424 0.408 0.392 0-377 0.361 0.344 0.328 0.312 0-295 0.278 0.261 0.244 0-227 0.210 0.192 0.175 0.157 0-139 0"121 0.103 0.084 0.066 0"047 0"028 0-010

1.073 1"134 1.180 1.219 1-255 1.290 1-323 1.355 1.384 1.415 1.444 1"473 1.502 1-531 1-560 1.588 1.617 1-646 1.675 1-702 1-732 1"760 1.790 1.820 1.848 1.879 1.909 1.939 1"969 2"000 2"031

u+

1

0.010 0.028 0-047 0.066 0*084 0.103 0-121 0-139 0.157 0.175 0.192 0.210 0.227 0.244 0"261 0.278 0-295 0.312 0-328 0-344 0-361 0"377 0"392 0.408 0.424 0-439 0.454 0-469 0"484 0-499 0"514

264

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value*; the value of dissociation energy recommended b y Hildebrand and Murad [14], however, would lead to an irregular behaviour of the RPC which could only be due to some perturbation. The RPC of P2, which could be calculated up to (u ~- 1) ---- 0.514, is given in Table 2. CONCLUSION The s t u d y of the ground state RPC's of diatomic molecules confirms the validity of the law leading to the RPC scheme indicated above. This suggests several applications of the RPC m e t h o d [5]. To name the most important: estimation of molecular constants, evaluation of the merits of various interatomic potentials, detection of experimental errors and nonadiabatic perturbations, and above all construction of highly reliable adiabatic potential curves, from R K R V curves of adjacent molecules or from theoretical ab initio calculated potential curves, for molecules the R K R V curves of which cannot be calculated or can be constructed only in a narrow neighbourhood of the equilibrium internuclear distance. Examples of some actual applications of the RPC method will be presented in further communications.

Acbnow[edgements--I thank Prof. J. PL~VA for his kind interest in this work and Mrs. ZAHRADNIKOVAfor technical help in preparing the manuscript. * It seems likely that the value of De(BoF) could in fact be somewhat lower; unfortunately no RPC of any adjacent molecule is available which could permit a more accurate check on

D~(BeF).