RKR Potentials of Isotopologues of the CO Molecule

ISSN 0030-400X, Optics and Spectroscopy, 015, Vol. 118, No. 1, pp. 6 10. Pleiades Publishing, Ltd., 015. Original Russian Text T.I. Velichko, S.N. Mikhailenko, 015, published in Optika i Spektroskopiya, 015, Vol. 118, No. 1, pp. 8 1. SPECTROSCOPY OF ATOMS AND MOLECULES RKR Potentials of Isotopologues of the CO Molecule T. I. Velichko a and S. N. Mikhailenko b, c a Uniersity of Architecture and Ciil Engineering, Tyumen, 65001 Russia b Zue Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences, Tomsk, 63401 Russia c National Research Tomsk Polytechnic Uniersity, Tomsk, 634050 Russia e-mail: tiel@list.ru, semen@iao.ru Receied May 30, 014 Abstract The RKR potentials of nine isotopologues of the carbon monoxide molecule in the ground electronic state are calculated on the basis of new alues of the Dunham spectroscopic parameters Y mj up to the = 41 ibrational leel. The potential of the main isotopologue 1 C 16 O determined at separate points is approximated by expansions in ariables zd = ( r re)/ re and zs = ( r re) / r. The expansion coefficients are presented. DOI: 10.1134/S0030400X1501045 INTRODUCTION The parameters of an intramolecular potential are required for numerical solution of the roibrational Schrödinger equation. The Rydberg Klein Rees (RKR) method, which is well known in spectroscopy of diatomics, allows the potential to be constructed point by point from the experimental alues of spectroscopic parameters; i.e., it allows one to calculate the minimum r min and maximum r max internuclear distances (the turning points) in states with a definite alue of ibrational quantum number ν. Earlier, the RKR potential of the main isotopologue 1 C 16 O was determined for 8 ibrational leels in [1] and for 37 leels in []. Subsequently [3], the RKR potentials of the isotopologues 1 C 16 O, 13 C 16 O, 1 C 17 O, 1 C 18 O, and 13 C 18 O calculated using the Dunham parameters [4] were reported for ibrational states up to = 40. In this paper, the RKR potentials were determined for nine isotopologues of the CO molecule in the ground electronic state. The calculations were carried out on the basis of the spectroscopic Dunham parameters Y mj deried recently [5]. Analytic approximations of the 1 C 16 O potential by expansions in different ariables were obtained. The choice of ariables and the region of applicability of the approximations obtained are discussed. CALCULATION OF RKR POTENTIAL The expressions for the turning points of a nuclear motion are written in the form [1, ] 1/ f f rmin( ) = f + f, rmax( ) = f + + f, g g 1/ where f = C min d' G( ) G( ' ) min and 1 B( ' ) d' g =.(1) C G( ) G( ' ) C = h, h is Planck s constant, c is the speed of 8π cμ light in a acuum, μ is the reduced mass of a molecule, G() is the rotational term, and B() is the rotational constant, which are written in terms of the Dunham coefficients Y mj : G( ) = Y ( + 0.5) + Y ( + 0.5) +, 10 0 B( ) = Y01 + Y11( + 0.5) + Y1( + 0.5) +. Methods for calculating integrals (1) containing a singularity at ' = are reiewed in [6]. In the present paper, the functions f and g are calculated using the approach deeloped in [1]. In [1], after integration by parts, the differences G() G(') were transferred to the numerator of the integrand. Equations (5) and (6) of [1] allowed us to easily calculate the functions f and g by numerical integration. The lower integration limit in (1) min = 0.5 Y 00 /Y 10 follows from the condition that ibrational energy E 0 = Y 00 + G() be zero for = min. Coefficient Y 00 was calculated from the approximate formula, Y11Y10 Y00 = 0.5 Y0 + Y01 1. 6Y 01 The alues of the potential in ibrational states with = 0, 1,,, 41 and corresponding turning points r min and r max were calculated for nine isotopologues of the CO molecule: 1 C 16 O, 13 C 16 O, 14 C 16 O, 1 C 17 O, 6

RKR POTENTIALS OF ISOTOPOLOGUES OF THE CO MOLECULE 7 Table 1. Vibrational energy leels and the turning points of the 13 C 16 O isotopologue E 0, сm 1 r min, Å r max, Å E 0, cm 1 r min, Å r max, Å 0 1057.77 1.0837807 1.1781691 1 39849.5857 0.9099873 1.604769 1 3153.7941 1.0541780 1.184789 4148.7959 0.9067016 1.617567 54.5463 1.0351457 1.483340 3 4984.341 0.903551 1.6388355 3 770.0453 1.004399 1.739809 4 44515.9758 0.9005300 1.6559775 4 990.354 1.008534 1.9791 5 4604.0948 0.897671 1.6731961 5 1185.5374 0.997766 1.3189047 6 47508.666 0.8948363 1.6905044 6 1355.6613 0.9885107 1.339467 7 48969.748 0.891510 1.7079148 7 1500.7934 0.98013 1.359035 8 50407.4179 0.8895653 1.754398 8 1711.0030 0.97679 1.3783075 9 5181.734 0.8870738 1.7430918 9 19016.3608 0.9657731 1.396917 30 531.7560 0.8846716 1.760889 10 0886.9390 0.9593948 1.4151359 31 54580.5376 0.88354 1.778854 11 73.8110 0.9534678 1.433048 3 5595.187 0.8801175 1.796931 1 4554.0517 0.947930 1.4507005 33 5746.5730 0.8779578 1.815156 13 6350.7369 0.947397 1.4681610 34 58544.9086 0.8758716 1.8336893 14 81.9435 0.9378519 1.4854661 35 5980.1666 0.8738557 1.853668 15 9870.7493 0.933361 1.506516 36 6107.3708 0.8719071 1.87165 16 31594.38 0.988653 1.5197480 37 6301.5368 0.870033 1.8903914 17 3393.4730 0.947166 1.5367819 38 63507.6715 0.868015 1.9097695 18 34968.549 0.907703 1.5537767 39 64690.7719 0.8664396 1.994135 19 36619.5406 0.9170096 1.5707536 40 65850.846 0.8647354 1.9493416 0 3846.566 0.9134196 1.587731 41 66987.8047 0.8630868 1.9695730 13 C 17 O, 14 C 17 O, 1 C 18 O, 13 C 18 O, and 14 C 18 O. The calculated results for two isotopologues 13 C 16 O and 1 C 18 O are listed in Tables 1 and. The results for other isotopologues are aailable from us upon request. The RKR potentials of 1 C 16 O, 13 C 17 O, and 14 C 18 O presented in the figure show the shift of the turning points on increasing the mass of an isotopologue. This shift of the turning points determines the changes in the energy structure of ibrational leels of different isotopologues of the molecule. ANALYTIC REPRESENTATION OF THE POTENTIAL BY THE DUNHAM AND SIMONS PARR FINLAN EXPANSIONS In order to obtain an analytic potential of the main isotopologue 1 C 16 O, its RKRs were approximated by the expansion i = 0 ( 1 + i ) U a z a z, () with the use of ariable z of two types: (i) Dunham ariable (D) z D = (r r e )/r e [7] and (ii) ariable of Simons Parr Finlan (SPF) z S = (r r e )/r [8], where r e is the equilibrium internuclear distance. a 0 has a dimension of cm 1, while all other parameters a i are dimensionless. To obtain the best approximation of the potential, the sum of squares of deiations between ibrational energies E 0(RKR) and the alues of U at the turning points was minimized by arying the parameters a i and equilibrium internuclear distance r e. The quality of fitting was characterized by rootmean-square deiation RMS = 1 ( E0( RKR) E0( calc) ) and relatie root-meansquare deiation RRMS N = 1 E 0( RKR) E 0( calc), where N is the number N E0( RKR) of potential points taken into account in the fitting procedure and E 0(calc) is the alue of U at the corresponding turning points. Approximations () were constructed for different sets of the sought parameters a i and different numbers of the RKRs being approximated. Table 3 presents the characteristics of fitting procedures inoling the points corresponding to 11 ibrational leels ( = 0, 1,,, 10; fittings 1 and ), 1 leels ( = 0, 1,,, 0; fittings 3 and 4), and 4 leels OPTICS AND SPECTROSCOPY Vol. 118 No. 1 015

8 VELICHKO, MIKHAILENKO Table. Vibrational energy leels and the turning points of the 1 C 18 O isotopologue E 0, сm 1 r min, Å r max, Å E 0, сm 1 r min, Å r max, Å 0 1055.7177 1.083806 1.1781183 1 39784.3884 0.910150 1.604080 1 3147.8393 1.054419 1.183833 41361.5857 0.9068396 1.61036 514.744 1.03538 1.48031 3 4915.0983 0.9036904 1.6380674 3 756.488 1.0058 1.738181 4 44445.001 0.9006684 1.6551735 4 973.1393 1.0083497 1.970363 5 45951.3678 0.8977655 1.673553 5 1164.7599 0.9978654 1.318689 6 47434.694 0.8949747 1.689658 6 1331.4157 0.9886189 1.339169 7 48893.7748 0.8989 1.7069974 7 15173.174 0.980360 1.358951 8 5039.9500 0.889703 1.74486 8 17090.1040 0.977959 1.3780010 9 5174.8573 0.887114 1.740935 9 1898.755 0.965893 1.396588 30 5313.5555 0.8848088 1.759844 10 0849.7604 0.9595178 1.4147733 31 54499.0991 0.884909 1.7777415 11 69.6317 0.9535933 1.436518 3 5584.5373 0.880537 1.795803 1 4510.9637 0.9480596 1.450809 33 5716.9144 0.8780933 1.8140404 13 6304.8319 0.948693 1.467714 34 58460.684 0.8760064 1.834661 14 8074.317 0.937983 1.4849881 35 59734.6308 0.8739898 1.8510939 15 9819.4834 0.9333688 1.501437 36 60986.057 0.870404 1.869938 16 31540.41 0.989993 1.519098 37 614.4693 0.8701556 1.8890137 17 3337.07 0.948516 1.53618 38 63419.9689 0.868339 1.908336 18 34909.9176 0.90906 1.553176 39 6460.54 0.8665700 1.9795 19 36558.63 0.9171463 1.570110 40 6576.1173 0.8648647 1.9477903 0 38183.497 0.9135568 1.5870658 41 66898.797 0.863149 1.9679587 Table 3. Characteristics of the fittings of RKR-points of 1 C 16 O by the D and SPF expansions Number D SPF Fitting no. Fitting no. of leels RMS, cm 1 RRMS RMS, сm 1 RRMS 11 ( points) 1 1.6 10 5 7.51 10 9 4.69 10 6.14 10 9 10 parameters 10 parameters 1 (4 points) 3.1 10 3 4.68 10 7 4 1.70 10 3 1.9 10 7 10 parameters 9 parameters 4 (84 points) 5 Solution is absent 6 7.3 10 3 1.66 10 6 8 parameters, a 6 was excluded from fitting Table 4. D and SPF parameters obtained in fittings 1 and to RKR-points D SPF D SPF r e, Å 1.18331834(46) 1.18331901(14) a 4 7.09(13) 0.677(58) a 0, сm 1 609451.34(73) 609451.347(1) a 5 7.657(11) 0.6714(9) a 1.6971630(1) 0.69715741(76) a 6 7.671(1) 0.778(19) a 4.506450() 0.5850546(76) a 7 6.99(1) 0.800(39) a 3 5.97107(3) 0.19047(74) a 8 4.35(41) 3.9(3) OPTICS AND SPECTROSCOPY Vol. 118 No. 1 015

RKR POTENTIALS OF ISOTOPOLOGUES OF THE CO MOLECULE 9 Energy, сm 1 60000 40000 0000 0 0.8 1.0 1. 1.4 1.6 1.8.0 Internuclear distance, Å RKR-potentials of 1 C 16 O, 13 C 17 O, and 14 C 18 O. The potentials of 13 C 17 O and 14 C 18 O are shifted by 0.05 and 0.050 Å, respectiely, for conenience. ( = 0, 1,,, 41; fittings 5 and 6). The lower part of the potential cure (11 ibrational leels) is reproduced most exactly by using both the D parameters and the SPF parameters (RMS = 1.6 10 5 and 4.69 10 6 cm 1, respectiely). It is known from the literature that the representation of a potential by a polynomial of r does not proide a high-precision description of highly excited ibrational states (see, for example, [9, 10]). One can see in Table 3 that the use of expansion () in the SPF ariable for approximating the 1 C 16 O potential is preferable. Moreoer, we failed to sole the fitting problem for all 84 RKRs (4 ibrational leels) by using the expansion in the Dunham ariable. The D and SPF parameters deried by fitting to 11 ibrational leels (fittings 1 and ) are listed in Table 4. One can see from the table that all adjustable Table 5. SPF parameters obtained in fitting 6 to 4 ibrational leels r e, Å 1.183565(67) a 0, сm 1 609467.8(35) a 1 0.696595(68) a 0.58776(13) a 3 0.1490(14) a 4 0.33764(97) a 5 0.966(10) a 6 0.0 a 7.360(3) parameters are statistically confident. Confidence interals σ in units of the last significant figures of the parameters are shown in parentheses. The SPF parameters obtained in fitting 6 upon approximating all 84 RKRs (4 ibrational leels) are listed in Table 5. Note that, in this fitting, coefficient a 6 was not statistically significant and was excluded from the treatment with a fixed zero alue (Table 5). The inaccuracy of calculations (E 0(approxim) E 0 ) of ibrational energies using the obtained approximants is rather small. For example, its alue is smaller than 0. cm 1 for the parameter alues deried from fitting 6 (Table 5). For other fittings, its alue does not exceed 0.004 cm 1 and is goerned by the accuracy of the obtained E 0(RKR) alues. CONCLUSIONS The RKR potentials determining the turning points of the nuclear motion in states with ibrational quantum number from 0 to 41, which corresponds to approximately 75% of the dissociation energy, were calculated for nine isotopologues of the CO molecule. The RKRs of the main 1 C 16 O isotopologue were approximated by power series expansions in ariables z D = (r r e )/r e and z S =(r r e )/r. The potential parameters obtained can be used to calculate both the spectroscopic characteristics of the molecule and the thermodynamic functions in modeling the physicochemical processes in carbon monoxide. The Dunham and Simons Parr Finlan parameters obtained from approximating the lower part of a potential cure ( 10) allow the potential to be calculated at the leel of accuracy of the experimental determination of spectral line centers ( 10 3 cm 1 ). OPTICS AND SPECTROSCOPY Vol. 118 No. 1 015

10 VELICHKO, MIKHAILENKO Howeer, all the obtained RKR potential points, including the highly excited ibrational states ( 41), can only be approximated by using the expansion in ariable z S (Tables 3 and 5). The inaccuracy of ibrational energy alues calculated with the use of parameters of Table 5 does not exceed 0. cm 1. ACKNOWLEDGMENTS This work was supported in part by the Russian Foundation for Basic Research (projects nos. 1-05- 93106-CNRS_a and 14-05-91150) and by the programs 3.9 and. of the Russian Academy of Sciences. REFERENCES 1. A. W. Mantz, J. K. G. Watson, K. N. Rao, D. L. Albritton, A. L. Schmeltekopf, and R. N. Zare, J. Mol. Spectrosc. 39 (1), 180 (1971).. S. M. Kirschner and J. K. G. Watson, J. Mol. Spectrosc. 47 (), 34 (1973). 3. C. Chackerian and D. Gooritch, NASA-TM-8466, 1 (198). 4. R. M. Dale, M. Herman, J. W. C. Johns, A. R. W. McKellar, S. Nagler, and I. K. M. Strathy, Can. J. Phys. 57 (5), 677 (1979). 5. T. I. Velichko, S. N. Mikhailenko, and S. A. Tashkun, J. Quant. Spectrosc. & Radiat. Transfer 113 (13), 1643 (01). 6. S. Chandra, A. K. Sharma, and Z. H. Khan, Pramana J. Phys. 47 (1), 65 (1996). 7. J. L. Dunham, Phys. Re. 41 (6), 71 (193). 8. G. Simons, R. G. Parr, and J. M. Finlan, J. Chem. Phys. 59 (6), 39 (1973). 9. A. V. Burenin and M. Yu. Ryabikin, Opt. Spektrosk. 78 (5), 667 (1995). 10. T. I. Velichko and Vl. G. Tyutere, Russian Phys. J. 11, 3 (1983). Translated by V. Bulyche OPTICS AND SPECTROSCOPY Vol. 118 No. 1 015