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Accurate potential energy curves for the X 1Σ+ and A1Σ+ states of NaH
41
Chemical Physics 121 (1988) 41-53 North-Holland, Amsterdam
ACCURATE A. PARDO, Departamento Received
POTENTIAL J.J. CAMACHO,
ENERGY CURVES J.M.L.
POYATO
de Quimica, Facultad de Cienciar C-XIV,
FOR THE X ‘2 + AND A ‘2 + STATES OF NaH and E. MARTIN
Universidad Authoma
de Madrid,
Cantoblanco,
28049 Madrid, Spain
28 July 1987
A reanalysis of the spectroscopic data for the ground and excited states of NaH yields improved spectroscopic constants (Dunham coefficients) which are used to compute new PMO-RKR-van der Waals potentials. The vibrational wavefunctions appropriate to the potential were used to obtain the eigenvalues E,, the rotational constant B, and the centrifugal distortion terms D,, H, and L,. They are in agreement with the experimental results within the accuracy of the measurements, as would be expected. Probability density distributions and Franck-Condon factors are calculated for the A’Z+ tf X ix+ band system of NaH. It is found that the probability distribution functions for the A’Z+ state show little asymmetry (as in harmonic oscillator) rather than the typical asymmetry favoring larger distances.
1. Introduction The A ‘2+ state of the alkali metal hydrides is of considerable interest because of the exceptional form of its potential energy curve. It is observed that AG( u + l/2) and B, for the upper state increase for small u before beginning to show their normal decreasing behavior with increasing u. This anomalous behavior of the A ‘C+ state is a consequence of the negative values of the constant w,x, (= - Y,,) and (Y, (= - Y,,). Therefore the potential of the A ‘2+ state, at first relatively harmonic (or even with an inverted anharmonicity [l]) becames extremely anharmonic. In this paper we report hybrid PMO-RKR-van der Waals potential energy curves for the A ix+ and X ‘2+ states of NaH. This method has also been used in the study of the different isotopic species of the lithium hydride [2]. From the experimental data presented in refs. [3-81, a revised set of spectroscopic constants for both the upper and lower states has been determined. Several multiconfiguration self-consistent-field (MC SCF) calculation have been performed for the lowest states of NaH [9,10]. Meyer and Rosmus [ll] performed a study on the ground state of NaH using their coupled electron pair approach (CEPA). Most recently, Olson and Liu [12] have published configuration-interaction (CI) potentials for the low-lying states of NaH and NaH-. The major reason for undertaking this work was to obtain accurate potential energy curves. The calculated potential has been checked by numerical solution of the radial wave equation. Good agreement is observed between the calculated eigenvalues and experimental term values. The rotational constants B, and the centrifugal distortion constants D,, H, and L, are computed from the vibrational eigenfunctions of the PMO-RKR-van der Waals rotationless potentials. From a practical standpoint, it is convenient to summarize adequately the hybrid PMO-RKR-van der Waals potentials by some simple analytical function. Thus, we have used Pad&approximant type functions where parameters are obtained by least-squares method. We also used the obtained wavefunctions to calculate the probability density distributions and the Franck-Condon factors for the A’Z+ ++ X ‘2+ band system of the NaH. It can be seen that the probability density distributions for the lowest vibrational levels of the A’Zf state have an inverted anharmonicity. 0301-0104/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
A. Pardo et al. / Potential energy curoes for the X
42
’ 2 + and A ’ .Z + states of NaH
2. Molecular constants The absorption spectrum of sodium hydride corresponding to the A ‘2+ * X ‘2+ transition was first studied by Hori [3]. Later this author enlarged the analysis by studying emission [4]. In these works some mistakes in the assigned quantum numbers indicated by Olson [5] are observed. These mistakes are clearly detected through simultaneous study of the deuteride species NaD. Afterwards, Pankhurst [6] made a study in emission extending the previous analysis from u’ = 1 to u’ = 20 and from u” = 0 to u” = 8. Much more recently, Orth and co-workers [7] have analyzed the A ‘2+ * X ‘2+ system, observing for the first time transitions to the vibrational level u’ = 0. This fact is due to the differences between the internuclear distances of the ground and excited state which, in the alkali hydrides, are very large. This means that generally only vibrational bands with high u’ are detected. The laser-induced fluorescence technique has also been used in the study of the sodium hydride [8]. In this study the vibrational information on the ground state is enlarged, with observation of vibrational levels up to u” = 15. Sastry et al. [13] have published an interesting millimeter wave spectroscopy study of NaH and NaD obtaining accurate rotational constants for the lowest observed vibrational states (u” = O-3) of the ground state. We have made a full reanalysis of earlier spectroscopic [3-81 data. From the wave numbers of the band origins of the NaH and NaD bands, corresponding to the transitions between two vibrational levels of the A ‘E+ and X ‘Z+ electronic states, we have made a least-squares fit to the expression:
vi= T,+ G(r/‘)
(1)
- G(q”),
where T, is the electronic energy and q’ and n” are the mass-reduced vibrational quantum numbers of the upper and the lower state, respectively. In eq. (l), 1) = (u + 1/2)/p and vi = Y( u’, u”)/p, where p = 1 for NaD and p = [p( 23NaH)/p( “NaD)] ‘I2 = 0 .722057246 for NaH. From this equation we obtain T, and the isotopically combined vibrational molecular constants for both electronic states (A-X transition) of NaD, and by using Dunham isotope relations, the vibrational molecular constants for NaH. The vibrational energy, in terms of Dunham coefficients, can be written as
G(v) = The rotational
Bd=
c Y,ovk.
(2)
k=l
constant
c Yklqk,
k=O
B in the vibrational
state u is given by (3)
where BL = B, for NaD and Bu = B,p2 for NaH. For the study of the ground state we have used, in particular, the Giroud and Nedelec analysis [8] because it contains more information for that electronic state. The considered experimental results for the ground state are restricted to u” = 15, although these values exist up to u” = 19 [14]. There is no information for the u” = 16 vibrational level. We do not retain these experimental values for u” > 17 because, as Giroud and Nedelec indicated, for u” > 15 the vibrational spacings and the rotational constants decrease very rapidly (possibly due to a sudden change of configuration mixing) and no satisfactory polynomial fit can be obtained. In table 1 the calculated isotopically combined molecular constants for the X ‘2+ and A ‘Z+ electronic states of NaH are presented.
A. Pardo et al. / Potential energy curues for the X ‘Z+ and A ‘CZ+ states of NaH
3. The A ‘2 + and X ‘Z + PMO-RKR-van
43
der Waals potentials
The potential energy curves of the X and A states of NaH were constructed by the PMO-RKR-van der Waals method previously described in ref. [2]. For input, we used the explicit formulas for G(U) and B, in table 1. The corresponding X ‘Z+ and A ‘Z+ potentials for the RKR region are shown in tables 2 and 3. In table 4 we show the perturbed Morse parameters b, of the PM0 potential for the A’Z+ and X ‘Z+ states employed around the equilibrium internuclear distance (region of the minimum) U,(r)=D,{[1-exp(-aq)]2+b~[l-exp(-aq)]4+...+b,,[l-exp(-aq)]12}, where q = r - r,, as well as the C, coefficients turning point) = Co + C6/r6 + C,/r* + C,,/r’O
U,(r)
(4)
of the expression
(large distance
region outside
the last RKR
+ C,,/r’*.
Table 1 Isotopically combined Dunham coefficients Tk (in cm-‘) of NaH (the uncertain figures quoted are in parentheses). The figure after E gives the respective power of ten. The numbers of figures retained are required to provide an accurate representation for the highest vibrational levels l$
i
(X lx+ state)
5; (A’Z+
k=O 0 1 2 3 4 5 6 7
k=l
k=O
4.901(99345146) -0.1460(1433170) 6.309(667235)E-3 - 9.39(0989088)E-4 6.1(77350944)E-5 - 1,(619155541)E-6
1171.40(090933) - 19.58(24527477) 0.174(69301639) -0.0110(5560362) 64(99284244)E-4 - 2.(742116134)E-5
Table 2 Potential energy curve (RKR region) and eigenvalues (in cm-‘) U
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Experimental
G(u)+ Yoo
4
581.80 1714.56 2809.45 3867.12 4888.10 5862.78 6821.44 7734.23 8611.12 9451.83 10255.77 11021.93 11748.76 12434.02 13074.67 13666.62
4.830450 4.694300 4.563975 4.436392 4.309468 4.181917 4.053066 3.922654 3.790640 3.657007 3.521570 3.383783 3.242538 3.095978 2.941299 2.774557
state) k=l 1.716(49106241) 0.0865(2828930) -0.olllq745222) 5,591(4538OO)E-4 - 1.34(5488007)E-5 l.O(84179854)E-7
316.723(904504) 2.719(91280997) 0.3395(5454623) - 0.0570(1598053) 3.2(76121863)E-3 - 9.3(48844157)E-5 l.(093363942)E-7
for the X ‘Z+ state of NaH. r:’ = 1.887485 A and Y, = 0.979 cm-’
r+ (A)
r- (A)
2.078397 2.244432 2.373779 2.489420 2.598182 2.703297 2.806672 2.909602 3.013089 3.118023 3.225312 3.336008 3.451450 3.573455 3.704594 3.848646
1.730665 1.633624 1.573938 1.529269 1.493332 1.463326 1.437720 1.415554 1.396162 1.379044 1.363801 1.350104 1.337674 1.326274 1.315704 1.305789
Eigenvalues
G(u)+ Yoo
4
581.74 1714.59 2809.56 3867.24 4888.56 5873.44 6822.25 7734.96 8611.52 9451.69 10254.81 11021.48 11747.76 12432.59 13072.61 13664.46
4.830657 4.694904 4.563909 4.435530 4.307794 4.180149 4.051802 3.922681 3.792213 3.660712 3.527909 3.392445 3.252903 3.106839 2.950743 2.779671
A. Pardo et al. / Potential energy curves for the X ‘.Z+ and A ‘.Z+ states of NaH
44
Table 3 Potential energy curve (RKR region) and eigenvalues (in cm -‘) Experimental
V
G(u)+ 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
159.92 482.93 813.03 1150.25 1494.05 1843.49 2197.44 2554.68 2914.02 3274.33 3634.59 3933.87 4351.38 4706.41 5058.34 5406.62 5750.73 6090.26 6024.84 6754.25 7078.48
for the AIZ+
state of NaH. r,’ = 3.186715 A and Yoo = 0.843 cm-’ Eigenvalues
r+ (A)
r- (A)
3.512237 3.741997 3.902012 4.035807 4.155830 4.267731 4.374632 4.478433 4.580353 4.681199 4.781512 4.881669 4.981955 5.082587 5.183776 5.285735 5.388678 5.492805 5.598266 5.705081 5.813040
2.850685 2.606039 2.448788 2.331356 2.238479 2.162495 2.098808 2.044391 1.997136 1.955521 1.918418 1.884971 1.854534 1.826588 1.800742 1.776695 1.754225 1.733182 1.713500 1.695197 1.678396
G(v)+
Y, 1.757047 1.823112 1.871612 1.905285 1.926577 1.937658 1.940431 1.936549 1.927429 1.914258 1.898014 1.879476 1.859234 1.837707 1.815153 1.791682 1.767271 1.741775 1.714941 1.686420 1.655782
Y,
B,
159.96 483.29 813.77 1151.00 1494.81 1844.28 2198.20 2555.28 2914.62 3274.84 3634.94 3994.14 4351.58 4706.49 5058.37 5406.66 5750.87 6090.50 6425.25 6754.84 7079.01
1.756179 1.820125 1.868749 1.904315 1.925051 1.936270 1.940354 1.936302 1.927492 1.914680 1.898912 1.879806 1.859892 1.838052 1.814809 1.790630 1.765751 1.739809 1.712575 1.685233 1.657822
The C,, values in eq. (5) and table 4 have nothing to do with the true long-range van. der Waals coefficients C, given, e.g., in ref. [15], which are estimated from a theoretical treatment. The C, coefficients (see eq. (5)) were obtained from least-squares fit to the last RKR outer turning points and taking into
Table 4 PM0 and long-range parameters for X ‘Z+ and A’P are in cm -‘2 , X isincm -’ and a in A-’
co
C, C. C 10 C,,
of NaH. The uncertain figures quoted are in parentheses. The C, coefficients
x ‘x+
A’P+
2.187543(49) 14648.0(59) 0.1028165(26) 0.2310930(98) - 0.1270814(53) -0.1400223(96) 0.2563035(82) - 0.2269744(02) - 0.0629897(50) 0.4576382(77) - 0.3248550(57)
- 0.527811(72) 52353.4(04) 4.2199331(14) 58.756975(30) 50.972954(21) - 262.12277(84) 6085.6448(14) 14424.437(84) - 174851.75(25) 683460.35(19) 5670696.9(76)
15899.9(98) 5.33495(924)E+ - l.O9722(607)E + - 1.56496(086)E + 2,19313(366)E+
5 8 9 10
10137.0(10) - 9.90251(304)E + 5.25256(441)E+ - 7.49346(237)E+ 1.51510(993)E+
6 9 11 13
A. Pardo et al. / Potential energy curoes
for the X ‘2Y+ and A tZf
Table 5 Energy eigenvalues and rotational constants for 1000 and 2000 points and extrapolated values (A’X+ cm-’ u
0
1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20
45
states of NaH
state of NaH). All values are in
Rotational constants
Energy eigenvalues N=lOOO
N=2000
extrap.
N=lOOO
N=2000
extrap.
159.94 483.25 813.69 1150.92 1494.56 1843.90 2197.68 2554.58 2913.70 3273.69 3633.53 3992.45 4349.60 4704.2 5055.75 5403.71 5747.59 6086.88 6421.27 6750.52 7074.35
159.95 483.28 813.75 1151.03 1494.74 1844.18 2198.08 2555.11 2914.39 3274.55 3634.58 3993.72 4351.09 4705.91 5057.71 5405.92 5750.05 6089.59 6424.25 6153.76 7077.85
159.96 483.29 813.77 1151.07 1494.81 1844.27 2198.21 2555.29 2914.62 3274.84 3634.94 3994.14 4351.58 4706.49 5058.36 5406.66 5750.87 6090.50 6425.25 6754.84 7079.02
1.756183 1.820123 1.868726 1.904281 1.925031 1.936229 1.940312 1.936312 1.927481 1.914881 1.898996 1.879914 1.86002 1.83826 1.815071 1.790937 1.76611 1.740268 1.71311 1.684814 1.658481
1.75618 1.820125 1.86875 1.904307 1.925046 1.93626 1.940343 1.936305 1.921491 1.914865 1.898933 1.879833 1.859924 1.838104 1.814875 1.790706 1.765841 1.739924 1.712709 1.685378 1.657987
1.756179 1.820125 1.868759 1.904315 1.925051 1.93627 1.940354 1.936302 1.927492 1.91486 1.898912 1.789806 1.859892 1.838052 1.814809 1.79063 1.765751 1.739809 1.712575 1.685233 1.657822
account an asymptote corresponding to the experimental dissociation energy [15] and the “ab initio” points based on the Olson and Liu paper [12]. We have scaled these points to fit smoothly onto the RKR zone near the outermost turning points (u” = 15 and u’ = 20). Values of PM0 parameters were calculated from Huffaker’s formulas [16-181. For the experimental region the RKR turning points are calculated including high-order correction terms in the Dunham coefficients qk_ For checking the precision of those hybrid potentials we obtain the eigenvalues and eigenfunctions \k,,( I) by solving numerically the radial Schrbdinger equation (‘Z states)
JL’k,,(r)= [-W2,‘dr2+U,(r>+PJ(J+l)/r’]\k,,(r), where /3= A/4ncp, r is the internuclear distance and n is the reduced mass. The numerical method employed for this purpose has been developed in our laboratory [19]. Briefly, from a trial potential U,(r) defined in N points (for example, N = 1000) we obtain the trial eigenvalues and eigenfunctions for each vibrational level. Then, for the same range of internuclear distances, we determine the potential in N’ points (for example N’ = 2000) calculating the eigenvalues and eigenfunctions for each vibrational level. Afterwards we extrapolate by a previously checked parabolic function obtaining the eigenvalues for N -+ cc. As an example, we show in table 5 the obtained results for the A ‘ZS+ state of NaH. For comparison with other authors, table 6 gives the calculated AG( u + l/2) (= G(u + 1) - G(u)) values for the two states studied for NaH, along with some available theoretical and experimental data. For the ground state we agree with the spectroscopic analysis of Giroud and Nedelec [8] to 1 cm-‘, but when compared to the theoretical calculation of Olson and Liu [12] deviations less than 14 cm-’ are observed. For the A’Z+ state the agreement with the spectroscopic analysis of Orth et al. [7] is better.
46
A. Pardo et al. / Potential energy curues for the X ‘Z+ and A ‘.Y+ states of NaH
Table 6 Comparison of AG(u + l/2) ”
a) b, ‘) d,
and A’Z+
states of NaH. All values are in cm-r
x lx+ AG(u +1/2)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
values for the X ‘I+
1132.8 1095.0 1057.7 1021.3 984.9 948.8 912.7 876.5 840.2 803.1 766.7 726.3 684.8 640.0 591.9
A’Z+ =)
AG(o +1/2) 1134.0 1094.3 1057.9 1021.0 985.1 948.8 912.6 876.8 840.7 804.1 766.2 728.2 684.0 639.1 593.5
b,
AG(u +1/2)
=)
AG( u + l/2)
1132.6 1094.0 1055.8 1018.3 981.1 844.2 907.6 870.7 833.7 795.9 757.1 716.7 673.9 628.0 577.8
a)
323.3 330.5 337.2 343.8 349.5 353.9 357.1 359.3 360.2 360.1 359.2 357.4 354.9 351.9 348.3 344.2 339.6 334.8 329.6 324.2
AG(u +1/2)
d,
323.6 330.4 337.2 343.6 349.3 353.8 357.3 359.5 360.5 360.4 359.3 357.4 354.8 351.7 348.2 344.3 340.0 335.3 330.0 323.7
AG(o +1/2)
=)
325.4 331.7 338.7 345.3 350.9 355.1 358.1 359.8 360.9 360.9 359.8 357.8 355.0 351.7 348.0 343.7 339.0 333.9 328.1 321.7
Calculated eigenvalues from this work. Experimental values from ref. [8]. Calculated values from ref. [12]. Experimental values from ref. [7].
To provide further checks on the rotationless potential energy functions, U,(r), we have determined the rotational constant I?, and the centrifugal distortion terms D,, H, and L, from the wavefunctions 1u) = ‘k‘,,(r). These terms are calculated from the following equations [20,21] (differentials du, du, dt, _. . are implicit)
(u I l/r2 I uj2
D”=P2LuG(U)-G(u)’ (u 11/r* I u>(u11/r* I t>(t IV2
H”~p3~+u~+u [G(U)
L=
-P”l
G(u)][G(~)
I u>
-G(U)]
- P*&j
(u 11/r* I
u+u [G(u)
u>*
- G(u)]*’
(9)
(w/f-* lw4l/~* l~>wl/~‘I~>(~ IV2 I u> / J szu t+u uzu [G(u)- G(u)][G(t)-G(u)][G(s)-G(u)]
(uI l/r2 I u>(u IW2 I t>(t I l/r2 I u> +2P3B,j j f+u u+u [G(u)-G(u)]*[G(t)-G(u)]
-iP2D” j
u+u
(u I l/r2 I a’ [G(u)-G(u)]*'
(10)
41
A. Pardo et al. / Potential energy curves for the X ‘X + and A ‘C?+ states of NaH Table I Calculated values of rotational constants D,, H, and L,, and experimental values for the A’Z+ values are in cm-’ and the uncertainties quoted are given in parentheses X lx+ state I, V
lo‘%:’
0 1 2 3 4 5 6
3.40(14) 3.35(35) 3.31(89) 3.28(11) 3.2q69) 3.20(35) 3.16(06)
7
3.11(85)
8
3.08(65)
A’Xf
(talc.)
104L$’ (exp.)
3.31 3.13 3.29 3.25 3.11 3.05 3.18 3.04 3.28
=) a) a) a) b, a’ b, a’ b,
1OsH:’ (talc.) 2.10(5) 2.22(8) 2.2q8) 2.14(l) 2.03(4) 1.82(O) 1.67(2) 1.45(S)
1OsH:’ (exp.)
2.17 =) 1.13 a) 2.58 a) 2.10 =) 1.2 b, 0.69 a’ 1.9 b,
and X ‘Z+
states of NaH. All
1012Lr(calc.) -
1.(92) 2.(42) 2.(06) 2.(01)
state 1040:
0’
0 1
2.14(01) 2.28(72)
2 3 4 5 6 I 8 9 10 11
2.33(18) 2.29(32) 2.20(41) 2.12(91) 2.05(18) 1.96(21) 1.91(56) 1.87(06) 1.82(17) 1.79(93)
(talc.)
a) Values from ref. [6].
104D:’ (exp.)
108H; (talc.)
lO*H: (exp.)
lo’*L:
2.15 2.38 2.25 2.25 2.20 2.19 2.07 2.01 1.93
8.08(S) 7.30(8)
7.2 b, 6.9 b, 5.10 a) 3.85 a) 3.04 a) 2.87 a) 2.02 =) 1.5 a)
- 3.8(6) - 2.4(4)
b, b, a) a) a) a) =) =) a)
6.03(7) 3.94(l) 3.13(9) 2.54(2) 1.33(3)
(talc.)
b, Values from ref. [7].
In eqs. (8)-(10) the sums over vibrational levels include contributions from vibrational continuum states. However, the non-diagonal (U # u) elements (u 11/r’ ( u) d ecreases rapidly in magnitude as u - u increases, this contribution being largest when u = u f 1 (u is adjacent to u). Therefore, the contribution from continuum states is seldom needed for vibrational levels considerably below the dissociation limit. Table 7 lists the D,, H, and L, values for the X ‘Z+ and A ‘2+ states of NaH obtained from eqs. (Q-(10) along with the experimental values. The calculated values for these constants are in qualitative agreement with the ones observed experimentally. Employing Padt approximants [22,23], we have determined analytic expressions for the hybrid potentials which are handy and fit precisely the starting potentials. PadC’s rational approximations are widely used in calculations because they are generally more efficient than polynomial approximations. We make least-squares fits to the expressions PA[N,
N]=
a,,z2(1+
d,~+...+d,,_,z~-2)/(l+e,z+...+eNzN),
(11)
where taking z = (r - re)/r [24] gives the fits greater flexibility. To reproduce the correct asymptotic behavior it is important to use polynomials whose numerator and denominator have the same degree in z.
A. Pardo et al. / Potential energy curves for the X ‘X + and A ‘Zc
48
slates of NaH
Table 8 Coefficients of the Pad6 approximant for the X *Z+ state of the NaH molecule. All values are in cm-‘. a,, = 70097.7074385 coefficients of the upper polynomial (d,,
, d,,)
-121.605344143, 1453.61069106, 1348.46444176, - 11678.4114727, -4246.68590066, 1.74847873394, - 89.2049883236, 56411.5478647, - 19616.272102, -164771.211234, 216047.651351, 214688.425507, - 765096.689732, 242118.074564, 1099439.09495, - 1014251.38014, - 212687.475476, 369630.880084 coefficients of the lower polynomial (e,,
, es,,)
1.92704896307, -88.5372540775, - 138.18817548,1400.19937738,1703.20407605, - 10972.6415886, -9278.0901106, 52904.3259641, - 83457.4114099, 361849.541488, 404012.05476, - 646917.379133, - 1611772.74473, 28666.7143257, - 168198.531111, 1612785.39788, 2992519.55231, -3337882.67547, -1184747.6873, 1635443.15913 dissociation energy = 15900.00 standard deviation = 0.001607329 regression coefficient = 0.99999992
In our calculations we use polynomials estimated dissociation energy from
with
N = 20, where
good
agreement
is observed
D,=a,(l+d,+...+d,,)/(l+e,+...+e,,),
between
the
(12)
and the corresponding experimental values. Tables 8 and 9 list the coefficients of expression (11) as well as the calculated De values obtained from eq. (12) for the X ‘Zf and A’Z+ potentials of NaH. Another very interesting problem that arises in the study of alkali metal hydrides is the ionic-covalent configuration interaction and curve crossing problem. Fig. 1 shows the hybrid PMO-RKR-van der Waals potential for X and A states from this work. The peculiar shape of the A rZZ+ potential can be explained [25] from the avoided crossing at an internuclear distance r,. If we define the crossing distance as the point of minimum gap AU(r) = U,(r) - U,(r) between the A ‘Zf and X ‘2+ potentials, we determined r, = 4.04(6) A. The difference potential AU(r) = U,(r) - Ux( r ) near the avoided crossing region is plotted
Table 9 Coefficients of the Pade approximant for the A’Z+
state of the NaH molecule. All values are in cm-’
a,, = 14616.7666803 coefficients of the upper polynomial (d,,
. , d,,)
2.20272806596, -31.087002972, 17.3819949745, 188.3602031, -177.506605632, -495.781035993, 522.162314057, 574.999844005, - 876.105639281, - 31.625865138, 1514.32202243, 55.8422276853, - 1861.7776044, - 1282.72509195,786.360873444,2119.11594962, 1659.96615178, 489.514856096 coefficients of the lower polynomial (et,.
, e2c)
-0.261172431841, - 35.0083982418, 101.101787535, 147.670267568, - 832.179078155, 44.8273151559, - 6818.09759422, 3940.65013412, 9878.61963915, - 4852.53977482, - 1498.8957293, - 8801.20349752, 4424.19040465, - 3608.83206748, - 1613.89703702, 3400.7149273, 2874.04254021, 696.289445458 dissociation energy = 10136.96 standard deviation = 0.009266 regression coefficient = 0.99999908
3075.97351641, 4054.40767982,
49
A. Pardo et al. / Potential energy curues for the X ‘2+ and A ‘2 + states of NaH
40000
NaH
r
-
D ” = 15900 e
X’CC
R e ‘I= I
0
2
0
cm-’
1.887A I
r I
I
I
4
6
8
10
R(A)
Fig. 1. PMO-RKR-van
der Waals potential
energy curves (in cm-’
and A) for the A’Z*
and X ‘Z+ states of NaH.
fig. 2. Also of interest is the crossing between the ionic potential and the real potential of the A ‘2+ state. In this case the crossing distance is P-,’= 3.86(O) A. The ionic potential, choosing as zero point of the energy scale the minimum of the ground state, is given by
in
Ui,,( r) = 5352(2.519)
- 11615(0.97)/r,
(13)
are uncertain. The constant term in eq. (13) is where r is in A, Uion(r) in cm-’ and figures in parentheses obtained from ref. [26]; ionization potentiaI(Na) - electron affinity(H) + 0,“. In both cases the crossing distance occurs in a region between v’ = 2 and u’ = 4.
30000
-
25000 ; 5 2
20000
-
15000
-
I 3”
10000
0
I 1
2
3
R(A)
4
5
6
I 7
Fig. 2. The energy gap AU(r) = U,(r) - U,(r) between the A ‘E+ and X ‘Z+ potential curves near the avoided crossing region.
50
A. Pardo et al. / Potential energy curves for the X ‘2 + and A ‘2 + states of NaH
4. Probability density distribution and Franck-Condon
factors
From the eigenfunctions generated in the calculation we have determined distributions 1q”(r) 12 for some vibrational levels. In fig. 3 we present those
*
Fig. 3. Probability density distributions
for the lowest vibrational
R(A)
the probability density functions for the lowest
’
levels (u’ = O-5) of the X ‘2+ and A’Z+
states of NaH.
A. Pardo et al. / Potential energy curoes for the X ‘Z+ and A ‘E+ states of NaH
51
vibrational levels of the X ‘2+ and A ‘Z+ states of NaH. It is observed that the anomalous behavior of the A’Z+ state is clearly shown in the probability density functions of the lowest vibrational levels with a molecular behavior resembling more a harmonic oscillator than an anharmonic one. Even for the u’ = 1 level a changed in asymmetry is observed. These same characteristics have been observed in other hydrides as KH and RbH [27-301. For the X ‘2+ state, sharper probability density distributions than for the A ‘2+ state are obtained showing a situation of greater binding, with a normal behavior in the whole range of vibrational levels. From the obtained wavefunctions we have also determined the Franck-Condon factors for the A ‘2+ @ X ‘Zf system of NaH. In table 10 we show the numerical results obtained through rotationless
Table 10 Franck-Condon
u’
factors (FCF) (for u” = 0,.
,20) for the A’Z+ * X ‘Z+ system of NaH
FCF
0
7.85-6, 0.000117. 0.000782, 0.00322, 0.00935, 0.0209, 0.0381, 0.0586, 0.0788, 0.0947, 0.104, 0.105, 0.099, 0.0883, 0.0748, 0.0607, 0.0474, 0.0357, 0.026, 0.0185, 0.0128
1
0.0000937, 0.00114, 0.00616, 0.0199, 0.0441, 0.0724, 0.0923, 0.0928. 0.0737, 0.0443, 0.0172, 0.00203, 0.00145, 0.0123, 0.0287, 0.0446, 0.0562, 0.0618, 0.0616, 0.057, 0.0497
2
0.00058,0.00566,0.0236,0.0571,0.0888,0.0935,0.0647,0.0242, 0.0000983, 0.00328, 0.0132, 0.0256, 0.0369
3
0.00246, 0.0186, 0.0573, 0.0948, 0.0882, 0.0403, 0.00273, 0.0093, 0.0396, 0.0539, 0.0391, 0.0134, 0.000209, 0.00656, 0.0232, 0.0363, 0.0381, 0.0292, 0.0159, 0.00488, 0.0000975
4
0.00797, 0.0447, 0.0947, 0.0916, 0.0319, 0.0000604, 0.0282, 0.0556, 0.0391, 0.00742, 0.00209, 0.0225, 0.0396, 0.0355, 0.0172, 0.0024, 0.00134, 0.0117, 0.024, 0.0304, 0.0281
5
0.0209, 0.82, 0.105, 0.0397, 0.000405, 0.039, 0.0563, 0.0194, 0.000475, 0.0243, 0.0427, 0.0281, 0.0048, 0.00194, 0.0179, 0.0316, 0.0297, 0.0162, 0.00347, 0.00024, 0.00642
6
0.0453, 0.114, 0.0685, 0.000353, 0.0399, 0.0565, 0.00966, 0.00863, 0.0419, 0.0342, 0.00457, 0.00517, 0.0276, 0.0338, 0.0173, 0.0154, 0.00329, 0.0169, 0.0267, 0.0245, 0.0138
7
0.0829, 0.117, 0.0142, 0.0281, 0.0629, 0.00857, 0.0161, 0.048, 0.0188, 0.000941, 0.0265, 0.03555, 0.0128, 0.0000831, 0.0136, 0.0284, 0.0242, 0.00856, 0.0@00518, 0.00534, 0.0166
8
0.128, 0.077, 0.00468, 0.0707, 0.0169, 0.0164, 0.0508, 0.00995, 0.00972, 0.0394, 0.021, 1.6E-6, 0.0168, 0.0316, 0.0177, 0.00104, 0.00534, 0.0201, 0.0244, 0.0146, 0.00292
9
0.167, 0.0203, 0.0539, 0.0443, 0.00777, 0.0556,0.00833,0.0182, 0.0193, 0.0033, 0.00153, 0.0129, 0.0211
0.0012,0.00657,0.0288,0.0481,0.0519,0.0403,0.0219,0.00657,
0.0416, 0.0069, 0.00876, 0.0335, 0.0196, 0.000215, 0.0116, 0.0268,
10
0.182, 0.00168, 0.087, 0.000369, 0.0583, 0.0148, 0.0206, 0.0419, 0.00112, 0.0235, 0.0327, 0.00303, 0.01, 0.0294, 0.0166, 0.000275, 0.00887, 0.0227, 0.0189, 0.00511, 0.000205
11
0.162, 0.0519, 0.0445,0.0383,0.0376,0.0142,0.047, 0.0129, 0.000129, 0.00762, 0.0195, 0.0175
12
0.114, 0.132, 1.87E-7, 0.079, 0.00131, 0.0582, 0.000624, 0.0407, 0.0167, 0.00881, 0.0354, 0.00676, 0.00781, 0.0283, 0.0127, 0.000235, 0.0148, 0.0221, 0.0091, 2.44E-6, 0.00719
13
0.0606, 0.163, 0.0601, 0.0624, 0.00778, 0.0435, 0.0158, 0.0176, 0.0335, 0.0000238, 0.0253, 0.0215, 4.68E-6, 0.00454, 0.00255, 0.0166, 0.0178, 0.00555
14
0.0218, 0.118, 0.163, 0.0219, 0.0455, 0.0365, 0.0246, 0.0228, 0.0306, 0.00345, 0.035, 0.00597, 0.0109, 0.0269, 0.00506, 0.00489, 0.0212, 0.0135, 0.000335, 0.00648, 0.0167
15
0.00432, 0.046, 0.155, 0.163, 0.0112, 0.0523, 0.021, 0.0333, 0.0106, 0.0352, 0.0000224, 0.0281, 0.0136, 0.0027, 0.0233, 0.012, 0.000242, 0.0137, 0.0174, 0.00442, 0.0085
0.000121, 0.0343, 0.0236, 0.000925, 0.0281, 0.0228, 0.000443, 0.0123, 0.0258,
0.0168, 0.0229,
A. Pardo et al. / Potential energy curves for the X
52
‘2 + and A ‘2 + states of NaH
0.15
kj u.
O.lO_
0.05 -
n
-.-_
“M
1
23
4
5
6
7
8
9
IO
11
12
13
14
t5
Fig. 4. Franck-Condon
16
v’+l
overlap integrals gression.
for the u” = 0 pro-
potentials (J = 0). To explain qualitatively the band intensity distribution in absorption, the Franck-Condon factors (FCF) of the u” = 0 progression are represented in fig. 4. Due to the rapid oscillation of the wavefunctions, a slight shift of some thousandths of A in the potential affects appreciably the intensity distribution. The great number of data for (u’, u”) bands makes their graphic representation difficult, although to provide a qualitative idea of their structures these data have been represented in fig. 5. A typical intensity distribution of the alkali hydrides is observed, where the intensity factors, for low vibrational levels (both u’ and u”), are very small.
NaH A’1
Fig. 5. Threedimensional
graphic
of the Franck-Condon
-x1,.+_
factors
for the A ‘Z + tf X
’ E + band system of NaH.
A. Pardo et al. / Poteniial energy curves for the X
‘2 + and A ‘X
l
slates of NaH
53
Acknowledgement We gratefully acknowledge the support received from the “Comisi6n y TCcnica” (project No. 1203/84) for this research.
Asesora
de Investigacibn
References [l] [2] (31 [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
A. Pardo, J.M.L. Poyato and MS. Guijarro, Anal. Fis. 79 A (1983) 121. A. Pardo, J.J. &macho and J.M.L. Poyato, Chem. Phys. 108 (1986) 15. T. Hori, Z. Physik 62 (1930) 352. T. Hori, Z. Physik 71 (1931) 478. E. Olsson, Z. Physik 93 (1934) 206. R.C. Pankhurst, Proc. Phys. Sot. (London) A 62 (1949) 191. F.B. Orth, W.C. Stwalley, S.C. Yang and Y.K. Hsieh, J. Mol. Spectry. 79 (1980) 314. M. Giroud and 0. Nedelec, J. Chem. Phys. 73 (1980) 4151. E.S. Sachs, J. Hinze and N.H. Sabelh, J. Chem. Phys. 62 (1975) 3367, 3377. A.M. Karo, M.A. Gardner and J.R. Hiskes, J. Chem. Phys. 68 (1978) 1942. W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. R.E. Olson and B. Liu, J. Chem. Phys. 73 (1980) 2817. K.V.L.N. Sastry, E. Herbst and F.C. de Lucia, J. Chem. Phys. 75 (1981) 4753. 0. Nedelec and M. Giroud, J. Chem. Phys. 79 (1983) 2121. W.T. Zemke, R.E. Olson, K.K. Verma, W.C. Stwalley and B. Liu, J. Chem. Phys. 80 (1984) 356. J.N. Huffaker, J. Chem. Phys. 64 (1976) 3175, 4654. P.H. Dwivedi and J.N. Huffaker, J. Chem. Phys. 66 (1977) 1726. J.N. Huffaker, J. Chem. Phys. 72 (1980) 2601. J.J. Camacho, Ph.D. Thesis, Universidad Autonoma de Madrid, Spain (1986). A. Dalgamo, in: Quantum theory, Vol. 1. Elements, ed. D.R. Bates (Academic Press, New York, 1961) ch. 5. D.L. Albritton, W.J. Harrop, A.L. Schmeltekopf and R.N. Zare, J. Mol. Spectry. 46 (1973) 25. G.A. Baker and J.L. Gammel, The PadC approximant in theoretical physics (Academic Press, New York, 1970). A. Pardo, J.J. &macho and J.M.L. Poyato, Chem. Phys. Letters 131 (1986) 490. G. Simons, R.G. Parr and J.M. Finlan, J. Chem. Phys. 59 (1973) 3229. R.S. Mulliken, Phys. Rev. 50 (1936) 1017, 1028. R.C. Weast and M.J. Astle, eds., Handbook of chemistry and physics, 63th ed. (CRC Press, Boca Raton, 1982). A. Pardo, J.M.L. Poyato, M.S. Guijarro and J.I. Femandez-Alonso, J. Mol. Spectry. 97 (1983) 248. A. Pardo, J.M.L. Poyato and J.J. &macho, Spectrochim. Acta 43 A (1987) 679. A. Pardo, J.J. &macho, J.M.L. Poyato and E. Martin, Chem. Phys., 17 (1987) 149. A. Pardo, J.M.L. Poyato, J.J. Camacho and E. Martin, Spectrochim. Acta, to be published.
Cientifica
Chemical Physics 121 (1988) 41-53 North-Holland, Amsterdam
ACCURATE A. PARDO, Departamento Received
POTENTIAL J.J. CAMACHO,
ENERGY CURVES J.M.L.
POYATO
de Quimica, Facultad de Cienciar C-XIV,
FOR THE X ‘2 + AND A ‘2 + STATES OF NaH and E. MARTIN
Universidad Authoma
de Madrid,
Cantoblanco,
28049 Madrid, Spain
28 July 1987
A reanalysis of the spectroscopic data for the ground and excited states of NaH yields improved spectroscopic constants (Dunham coefficients) which are used to compute new PMO-RKR-van der Waals potentials. The vibrational wavefunctions appropriate to the potential were used to obtain the eigenvalues E,, the rotational constant B, and the centrifugal distortion terms D,, H, and L,. They are in agreement with the experimental results within the accuracy of the measurements, as would be expected. Probability density distributions and Franck-Condon factors are calculated for the A’Z+ tf X ix+ band system of NaH. It is found that the probability distribution functions for the A’Z+ state show little asymmetry (as in harmonic oscillator) rather than the typical asymmetry favoring larger distances.
1. Introduction The A ‘2+ state of the alkali metal hydrides is of considerable interest because of the exceptional form of its potential energy curve. It is observed that AG( u + l/2) and B, for the upper state increase for small u before beginning to show their normal decreasing behavior with increasing u. This anomalous behavior of the A ‘C+ state is a consequence of the negative values of the constant w,x, (= - Y,,) and (Y, (= - Y,,). Therefore the potential of the A ‘2+ state, at first relatively harmonic (or even with an inverted anharmonicity [l]) becames extremely anharmonic. In this paper we report hybrid PMO-RKR-van der Waals potential energy curves for the A ix+ and X ‘2+ states of NaH. This method has also been used in the study of the different isotopic species of the lithium hydride [2]. From the experimental data presented in refs. [3-81, a revised set of spectroscopic constants for both the upper and lower states has been determined. Several multiconfiguration self-consistent-field (MC SCF) calculation have been performed for the lowest states of NaH [9,10]. Meyer and Rosmus [ll] performed a study on the ground state of NaH using their coupled electron pair approach (CEPA). Most recently, Olson and Liu [12] have published configuration-interaction (CI) potentials for the low-lying states of NaH and NaH-. The major reason for undertaking this work was to obtain accurate potential energy curves. The calculated potential has been checked by numerical solution of the radial wave equation. Good agreement is observed between the calculated eigenvalues and experimental term values. The rotational constants B, and the centrifugal distortion constants D,, H, and L, are computed from the vibrational eigenfunctions of the PMO-RKR-van der Waals rotationless potentials. From a practical standpoint, it is convenient to summarize adequately the hybrid PMO-RKR-van der Waals potentials by some simple analytical function. Thus, we have used Pad&approximant type functions where parameters are obtained by least-squares method. We also used the obtained wavefunctions to calculate the probability density distributions and the Franck-Condon factors for the A’Z+ ++ X ‘2+ band system of the NaH. It can be seen that the probability density distributions for the lowest vibrational levels of the A’Zf state have an inverted anharmonicity. 0301-0104/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
A. Pardo et al. / Potential energy curoes for the X
42
’ 2 + and A ’ .Z + states of NaH
2. Molecular constants The absorption spectrum of sodium hydride corresponding to the A ‘2+ * X ‘2+ transition was first studied by Hori [3]. Later this author enlarged the analysis by studying emission [4]. In these works some mistakes in the assigned quantum numbers indicated by Olson [5] are observed. These mistakes are clearly detected through simultaneous study of the deuteride species NaD. Afterwards, Pankhurst [6] made a study in emission extending the previous analysis from u’ = 1 to u’ = 20 and from u” = 0 to u” = 8. Much more recently, Orth and co-workers [7] have analyzed the A ‘2+ * X ‘2+ system, observing for the first time transitions to the vibrational level u’ = 0. This fact is due to the differences between the internuclear distances of the ground and excited state which, in the alkali hydrides, are very large. This means that generally only vibrational bands with high u’ are detected. The laser-induced fluorescence technique has also been used in the study of the sodium hydride [8]. In this study the vibrational information on the ground state is enlarged, with observation of vibrational levels up to u” = 15. Sastry et al. [13] have published an interesting millimeter wave spectroscopy study of NaH and NaD obtaining accurate rotational constants for the lowest observed vibrational states (u” = O-3) of the ground state. We have made a full reanalysis of earlier spectroscopic [3-81 data. From the wave numbers of the band origins of the NaH and NaD bands, corresponding to the transitions between two vibrational levels of the A ‘E+ and X ‘Z+ electronic states, we have made a least-squares fit to the expression:
vi= T,+ G(r/‘)
(1)
- G(q”),
where T, is the electronic energy and q’ and n” are the mass-reduced vibrational quantum numbers of the upper and the lower state, respectively. In eq. (l), 1) = (u + 1/2)/p and vi = Y( u’, u”)/p, where p = 1 for NaD and p = [p( 23NaH)/p( “NaD)] ‘I2 = 0 .722057246 for NaH. From this equation we obtain T, and the isotopically combined vibrational molecular constants for both electronic states (A-X transition) of NaD, and by using Dunham isotope relations, the vibrational molecular constants for NaH. The vibrational energy, in terms of Dunham coefficients, can be written as
G(v) = The rotational
Bd=
c Y,ovk.
(2)
k=l
constant
c Yklqk,
k=O
B in the vibrational
state u is given by (3)
where BL = B, for NaD and Bu = B,p2 for NaH. For the study of the ground state we have used, in particular, the Giroud and Nedelec analysis [8] because it contains more information for that electronic state. The considered experimental results for the ground state are restricted to u” = 15, although these values exist up to u” = 19 [14]. There is no information for the u” = 16 vibrational level. We do not retain these experimental values for u” > 17 because, as Giroud and Nedelec indicated, for u” > 15 the vibrational spacings and the rotational constants decrease very rapidly (possibly due to a sudden change of configuration mixing) and no satisfactory polynomial fit can be obtained. In table 1 the calculated isotopically combined molecular constants for the X ‘2+ and A ‘Z+ electronic states of NaH are presented.
A. Pardo et al. / Potential energy curues for the X ‘Z+ and A ‘CZ+ states of NaH
3. The A ‘2 + and X ‘Z + PMO-RKR-van
43
der Waals potentials
The potential energy curves of the X and A states of NaH were constructed by the PMO-RKR-van der Waals method previously described in ref. [2]. For input, we used the explicit formulas for G(U) and B, in table 1. The corresponding X ‘Z+ and A ‘Z+ potentials for the RKR region are shown in tables 2 and 3. In table 4 we show the perturbed Morse parameters b, of the PM0 potential for the A’Z+ and X ‘Z+ states employed around the equilibrium internuclear distance (region of the minimum) U,(r)=D,{[1-exp(-aq)]2+b~[l-exp(-aq)]4+...+b,,[l-exp(-aq)]12}, where q = r - r,, as well as the C, coefficients turning point) = Co + C6/r6 + C,/r* + C,,/r’O
U,(r)
(4)
of the expression
(large distance
region outside
the last RKR
+ C,,/r’*.
Table 1 Isotopically combined Dunham coefficients Tk (in cm-‘) of NaH (the uncertain figures quoted are in parentheses). The figure after E gives the respective power of ten. The numbers of figures retained are required to provide an accurate representation for the highest vibrational levels l$
i
(X lx+ state)
5; (A’Z+
k=O 0 1 2 3 4 5 6 7
k=l
k=O
4.901(99345146) -0.1460(1433170) 6.309(667235)E-3 - 9.39(0989088)E-4 6.1(77350944)E-5 - 1,(619155541)E-6
1171.40(090933) - 19.58(24527477) 0.174(69301639) -0.0110(5560362) 64(99284244)E-4 - 2.(742116134)E-5
Table 2 Potential energy curve (RKR region) and eigenvalues (in cm-‘) U
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Experimental
G(u)+ Yoo
4
581.80 1714.56 2809.45 3867.12 4888.10 5862.78 6821.44 7734.23 8611.12 9451.83 10255.77 11021.93 11748.76 12434.02 13074.67 13666.62
4.830450 4.694300 4.563975 4.436392 4.309468 4.181917 4.053066 3.922654 3.790640 3.657007 3.521570 3.383783 3.242538 3.095978 2.941299 2.774557
state) k=l 1.716(49106241) 0.0865(2828930) -0.olllq745222) 5,591(4538OO)E-4 - 1.34(5488007)E-5 l.O(84179854)E-7
316.723(904504) 2.719(91280997) 0.3395(5454623) - 0.0570(1598053) 3.2(76121863)E-3 - 9.3(48844157)E-5 l.(093363942)E-7
for the X ‘Z+ state of NaH. r:’ = 1.887485 A and Y, = 0.979 cm-’
r+ (A)
r- (A)
2.078397 2.244432 2.373779 2.489420 2.598182 2.703297 2.806672 2.909602 3.013089 3.118023 3.225312 3.336008 3.451450 3.573455 3.704594 3.848646
1.730665 1.633624 1.573938 1.529269 1.493332 1.463326 1.437720 1.415554 1.396162 1.379044 1.363801 1.350104 1.337674 1.326274 1.315704 1.305789
Eigenvalues
G(u)+ Yoo
4
581.74 1714.59 2809.56 3867.24 4888.56 5873.44 6822.25 7734.96 8611.52 9451.69 10254.81 11021.48 11747.76 12432.59 13072.61 13664.46
4.830657 4.694904 4.563909 4.435530 4.307794 4.180149 4.051802 3.922681 3.792213 3.660712 3.527909 3.392445 3.252903 3.106839 2.950743 2.779671
A. Pardo et al. / Potential energy curves for the X ‘.Z+ and A ‘.Z+ states of NaH
44
Table 3 Potential energy curve (RKR region) and eigenvalues (in cm -‘) Experimental
V
G(u)+ 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
159.92 482.93 813.03 1150.25 1494.05 1843.49 2197.44 2554.68 2914.02 3274.33 3634.59 3933.87 4351.38 4706.41 5058.34 5406.62 5750.73 6090.26 6024.84 6754.25 7078.48
for the AIZ+
state of NaH. r,’ = 3.186715 A and Yoo = 0.843 cm-’ Eigenvalues
r+ (A)
r- (A)
3.512237 3.741997 3.902012 4.035807 4.155830 4.267731 4.374632 4.478433 4.580353 4.681199 4.781512 4.881669 4.981955 5.082587 5.183776 5.285735 5.388678 5.492805 5.598266 5.705081 5.813040
2.850685 2.606039 2.448788 2.331356 2.238479 2.162495 2.098808 2.044391 1.997136 1.955521 1.918418 1.884971 1.854534 1.826588 1.800742 1.776695 1.754225 1.733182 1.713500 1.695197 1.678396
G(v)+
Y, 1.757047 1.823112 1.871612 1.905285 1.926577 1.937658 1.940431 1.936549 1.927429 1.914258 1.898014 1.879476 1.859234 1.837707 1.815153 1.791682 1.767271 1.741775 1.714941 1.686420 1.655782
Y,
B,
159.96 483.29 813.77 1151.00 1494.81 1844.28 2198.20 2555.28 2914.62 3274.84 3634.94 3994.14 4351.58 4706.49 5058.37 5406.66 5750.87 6090.50 6425.25 6754.84 7079.01
1.756179 1.820125 1.868749 1.904315 1.925051 1.936270 1.940354 1.936302 1.927492 1.914680 1.898912 1.879806 1.859892 1.838052 1.814809 1.790630 1.765751 1.739809 1.712575 1.685233 1.657822
The C,, values in eq. (5) and table 4 have nothing to do with the true long-range van. der Waals coefficients C, given, e.g., in ref. [15], which are estimated from a theoretical treatment. The C, coefficients (see eq. (5)) were obtained from least-squares fit to the last RKR outer turning points and taking into
Table 4 PM0 and long-range parameters for X ‘Z+ and A’P are in cm -‘2 , X isincm -’ and a in A-’
co
C, C. C 10 C,,
of NaH. The uncertain figures quoted are in parentheses. The C, coefficients
x ‘x+
A’P+
2.187543(49) 14648.0(59) 0.1028165(26) 0.2310930(98) - 0.1270814(53) -0.1400223(96) 0.2563035(82) - 0.2269744(02) - 0.0629897(50) 0.4576382(77) - 0.3248550(57)
- 0.527811(72) 52353.4(04) 4.2199331(14) 58.756975(30) 50.972954(21) - 262.12277(84) 6085.6448(14) 14424.437(84) - 174851.75(25) 683460.35(19) 5670696.9(76)
15899.9(98) 5.33495(924)E+ - l.O9722(607)E + - 1.56496(086)E + 2,19313(366)E+
5 8 9 10
10137.0(10) - 9.90251(304)E + 5.25256(441)E+ - 7.49346(237)E+ 1.51510(993)E+
6 9 11 13
A. Pardo et al. / Potential energy curoes
for the X ‘2Y+ and A tZf
Table 5 Energy eigenvalues and rotational constants for 1000 and 2000 points and extrapolated values (A’X+ cm-’ u
0
1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20
45
states of NaH
state of NaH). All values are in
Rotational constants
Energy eigenvalues N=lOOO
N=2000
extrap.
N=lOOO
N=2000
extrap.
159.94 483.25 813.69 1150.92 1494.56 1843.90 2197.68 2554.58 2913.70 3273.69 3633.53 3992.45 4349.60 4704.2 5055.75 5403.71 5747.59 6086.88 6421.27 6750.52 7074.35
159.95 483.28 813.75 1151.03 1494.74 1844.18 2198.08 2555.11 2914.39 3274.55 3634.58 3993.72 4351.09 4705.91 5057.71 5405.92 5750.05 6089.59 6424.25 6153.76 7077.85
159.96 483.29 813.77 1151.07 1494.81 1844.27 2198.21 2555.29 2914.62 3274.84 3634.94 3994.14 4351.58 4706.49 5058.36 5406.66 5750.87 6090.50 6425.25 6754.84 7079.02
1.756183 1.820123 1.868726 1.904281 1.925031 1.936229 1.940312 1.936312 1.927481 1.914881 1.898996 1.879914 1.86002 1.83826 1.815071 1.790937 1.76611 1.740268 1.71311 1.684814 1.658481
1.75618 1.820125 1.86875 1.904307 1.925046 1.93626 1.940343 1.936305 1.921491 1.914865 1.898933 1.879833 1.859924 1.838104 1.814875 1.790706 1.765841 1.739924 1.712709 1.685378 1.657987
1.756179 1.820125 1.868759 1.904315 1.925051 1.93627 1.940354 1.936302 1.927492 1.91486 1.898912 1.789806 1.859892 1.838052 1.814809 1.79063 1.765751 1.739809 1.712575 1.685233 1.657822
account an asymptote corresponding to the experimental dissociation energy [15] and the “ab initio” points based on the Olson and Liu paper [12]. We have scaled these points to fit smoothly onto the RKR zone near the outermost turning points (u” = 15 and u’ = 20). Values of PM0 parameters were calculated from Huffaker’s formulas [16-181. For the experimental region the RKR turning points are calculated including high-order correction terms in the Dunham coefficients qk_ For checking the precision of those hybrid potentials we obtain the eigenvalues and eigenfunctions \k,,( I) by solving numerically the radial Schrbdinger equation (‘Z states)
JL’k,,(r)= [-W2,‘dr2+U,(r>+PJ(J+l)/r’]\k,,(r), where /3= A/4ncp, r is the internuclear distance and n is the reduced mass. The numerical method employed for this purpose has been developed in our laboratory [19]. Briefly, from a trial potential U,(r) defined in N points (for example, N = 1000) we obtain the trial eigenvalues and eigenfunctions for each vibrational level. Then, for the same range of internuclear distances, we determine the potential in N’ points (for example N’ = 2000) calculating the eigenvalues and eigenfunctions for each vibrational level. Afterwards we extrapolate by a previously checked parabolic function obtaining the eigenvalues for N -+ cc. As an example, we show in table 5 the obtained results for the A ‘ZS+ state of NaH. For comparison with other authors, table 6 gives the calculated AG( u + l/2) (= G(u + 1) - G(u)) values for the two states studied for NaH, along with some available theoretical and experimental data. For the ground state we agree with the spectroscopic analysis of Giroud and Nedelec [8] to 1 cm-‘, but when compared to the theoretical calculation of Olson and Liu [12] deviations less than 14 cm-’ are observed. For the A’Z+ state the agreement with the spectroscopic analysis of Orth et al. [7] is better.
46
A. Pardo et al. / Potential energy curues for the X ‘Z+ and A ‘.Y+ states of NaH
Table 6 Comparison of AG(u + l/2) ”
a) b, ‘) d,
and A’Z+
states of NaH. All values are in cm-r
x lx+ AG(u +1/2)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
values for the X ‘I+
1132.8 1095.0 1057.7 1021.3 984.9 948.8 912.7 876.5 840.2 803.1 766.7 726.3 684.8 640.0 591.9
A’Z+ =)
AG(o +1/2) 1134.0 1094.3 1057.9 1021.0 985.1 948.8 912.6 876.8 840.7 804.1 766.2 728.2 684.0 639.1 593.5
b,
AG(u +1/2)
=)
AG( u + l/2)
1132.6 1094.0 1055.8 1018.3 981.1 844.2 907.6 870.7 833.7 795.9 757.1 716.7 673.9 628.0 577.8
a)
323.3 330.5 337.2 343.8 349.5 353.9 357.1 359.3 360.2 360.1 359.2 357.4 354.9 351.9 348.3 344.2 339.6 334.8 329.6 324.2
AG(u +1/2)
d,
323.6 330.4 337.2 343.6 349.3 353.8 357.3 359.5 360.5 360.4 359.3 357.4 354.8 351.7 348.2 344.3 340.0 335.3 330.0 323.7
AG(o +1/2)
=)
325.4 331.7 338.7 345.3 350.9 355.1 358.1 359.8 360.9 360.9 359.8 357.8 355.0 351.7 348.0 343.7 339.0 333.9 328.1 321.7
Calculated eigenvalues from this work. Experimental values from ref. [8]. Calculated values from ref. [12]. Experimental values from ref. [7].
To provide further checks on the rotationless potential energy functions, U,(r), we have determined the rotational constant I?, and the centrifugal distortion terms D,, H, and L, from the wavefunctions 1u) = ‘k‘,,(r). These terms are calculated from the following equations [20,21] (differentials du, du, dt, _. . are implicit)
(u I l/r2 I uj2
D”=P2LuG(U)-G(u)’ (u 11/r* I u>(u11/r* I t>(t IV2
H”~p3~+u~+u [G(U)
L=
-P”l
G(u)][G(~)
I u>
-G(U)]
- P*&j
(u 11/r* I
u+u [G(u)
u>*
- G(u)]*’
(9)
(w/f-* lw4l/~* l~>wl/~‘I~>(~ IV2 I u> / J szu t+u uzu [G(u)- G(u)][G(t)-G(u)][G(s)-G(u)]
(uI l/r2 I u>(u IW2 I t>(t I l/r2 I u> +2P3B,j j f+u u+u [G(u)-G(u)]*[G(t)-G(u)]
-iP2D” j
u+u
(u I l/r2 I a’ [G(u)-G(u)]*'
(10)
41
A. Pardo et al. / Potential energy curves for the X ‘X + and A ‘C?+ states of NaH Table I Calculated values of rotational constants D,, H, and L,, and experimental values for the A’Z+ values are in cm-’ and the uncertainties quoted are given in parentheses X lx+ state I, V
lo‘%:’
0 1 2 3 4 5 6
3.40(14) 3.35(35) 3.31(89) 3.28(11) 3.2q69) 3.20(35) 3.16(06)
7
3.11(85)
8
3.08(65)
A’Xf
(talc.)
104L$’ (exp.)
3.31 3.13 3.29 3.25 3.11 3.05 3.18 3.04 3.28
=) a) a) a) b, a’ b, a’ b,
1OsH:’ (talc.) 2.10(5) 2.22(8) 2.2q8) 2.14(l) 2.03(4) 1.82(O) 1.67(2) 1.45(S)
1OsH:’ (exp.)
2.17 =) 1.13 a) 2.58 a) 2.10 =) 1.2 b, 0.69 a’ 1.9 b,
and X ‘Z+
states of NaH. All
1012Lr(calc.) -
1.(92) 2.(42) 2.(06) 2.(01)
state 1040:
0’
0 1
2.14(01) 2.28(72)
2 3 4 5 6 I 8 9 10 11
2.33(18) 2.29(32) 2.20(41) 2.12(91) 2.05(18) 1.96(21) 1.91(56) 1.87(06) 1.82(17) 1.79(93)
(talc.)
a) Values from ref. [6].
104D:’ (exp.)
108H; (talc.)
lO*H: (exp.)
lo’*L:
2.15 2.38 2.25 2.25 2.20 2.19 2.07 2.01 1.93
8.08(S) 7.30(8)
7.2 b, 6.9 b, 5.10 a) 3.85 a) 3.04 a) 2.87 a) 2.02 =) 1.5 a)
- 3.8(6) - 2.4(4)
b, b, a) a) a) a) =) =) a)
6.03(7) 3.94(l) 3.13(9) 2.54(2) 1.33(3)
(talc.)
b, Values from ref. [7].
In eqs. (8)-(10) the sums over vibrational levels include contributions from vibrational continuum states. However, the non-diagonal (U # u) elements (u 11/r’ ( u) d ecreases rapidly in magnitude as u - u increases, this contribution being largest when u = u f 1 (u is adjacent to u). Therefore, the contribution from continuum states is seldom needed for vibrational levels considerably below the dissociation limit. Table 7 lists the D,, H, and L, values for the X ‘Z+ and A ‘2+ states of NaH obtained from eqs. (Q-(10) along with the experimental values. The calculated values for these constants are in qualitative agreement with the ones observed experimentally. Employing Padt approximants [22,23], we have determined analytic expressions for the hybrid potentials which are handy and fit precisely the starting potentials. PadC’s rational approximations are widely used in calculations because they are generally more efficient than polynomial approximations. We make least-squares fits to the expressions PA[N,
N]=
a,,z2(1+
d,~+...+d,,_,z~-2)/(l+e,z+...+eNzN),
(11)
where taking z = (r - re)/r [24] gives the fits greater flexibility. To reproduce the correct asymptotic behavior it is important to use polynomials whose numerator and denominator have the same degree in z.
A. Pardo et al. / Potential energy curves for the X ‘X + and A ‘Zc
48
slates of NaH
Table 8 Coefficients of the Pad6 approximant for the X *Z+ state of the NaH molecule. All values are in cm-‘. a,, = 70097.7074385 coefficients of the upper polynomial (d,,
, d,,)
-121.605344143, 1453.61069106, 1348.46444176, - 11678.4114727, -4246.68590066, 1.74847873394, - 89.2049883236, 56411.5478647, - 19616.272102, -164771.211234, 216047.651351, 214688.425507, - 765096.689732, 242118.074564, 1099439.09495, - 1014251.38014, - 212687.475476, 369630.880084 coefficients of the lower polynomial (e,,
, es,,)
1.92704896307, -88.5372540775, - 138.18817548,1400.19937738,1703.20407605, - 10972.6415886, -9278.0901106, 52904.3259641, - 83457.4114099, 361849.541488, 404012.05476, - 646917.379133, - 1611772.74473, 28666.7143257, - 168198.531111, 1612785.39788, 2992519.55231, -3337882.67547, -1184747.6873, 1635443.15913 dissociation energy = 15900.00 standard deviation = 0.001607329 regression coefficient = 0.99999992
In our calculations we use polynomials estimated dissociation energy from
with
N = 20, where
good
agreement
is observed
D,=a,(l+d,+...+d,,)/(l+e,+...+e,,),
between
the
(12)
and the corresponding experimental values. Tables 8 and 9 list the coefficients of expression (11) as well as the calculated De values obtained from eq. (12) for the X ‘Zf and A’Z+ potentials of NaH. Another very interesting problem that arises in the study of alkali metal hydrides is the ionic-covalent configuration interaction and curve crossing problem. Fig. 1 shows the hybrid PMO-RKR-van der Waals potential for X and A states from this work. The peculiar shape of the A rZZ+ potential can be explained [25] from the avoided crossing at an internuclear distance r,. If we define the crossing distance as the point of minimum gap AU(r) = U,(r) - U,(r) between the A ‘Zf and X ‘2+ potentials, we determined r, = 4.04(6) A. The difference potential AU(r) = U,(r) - Ux( r ) near the avoided crossing region is plotted
Table 9 Coefficients of the Pade approximant for the A’Z+
state of the NaH molecule. All values are in cm-’
a,, = 14616.7666803 coefficients of the upper polynomial (d,,
. , d,,)
2.20272806596, -31.087002972, 17.3819949745, 188.3602031, -177.506605632, -495.781035993, 522.162314057, 574.999844005, - 876.105639281, - 31.625865138, 1514.32202243, 55.8422276853, - 1861.7776044, - 1282.72509195,786.360873444,2119.11594962, 1659.96615178, 489.514856096 coefficients of the lower polynomial (et,.
, e2c)
-0.261172431841, - 35.0083982418, 101.101787535, 147.670267568, - 832.179078155, 44.8273151559, - 6818.09759422, 3940.65013412, 9878.61963915, - 4852.53977482, - 1498.8957293, - 8801.20349752, 4424.19040465, - 3608.83206748, - 1613.89703702, 3400.7149273, 2874.04254021, 696.289445458 dissociation energy = 10136.96 standard deviation = 0.009266 regression coefficient = 0.99999908
3075.97351641, 4054.40767982,
49
A. Pardo et al. / Potential energy curues for the X ‘2+ and A ‘2 + states of NaH
40000
NaH
r
-
D ” = 15900 e
X’CC
R e ‘I= I
0
2
0
cm-’
1.887A I
r I
I
I
4
6
8
10
R(A)
Fig. 1. PMO-RKR-van
der Waals potential
energy curves (in cm-’
and A) for the A’Z*
and X ‘Z+ states of NaH.
fig. 2. Also of interest is the crossing between the ionic potential and the real potential of the A ‘2+ state. In this case the crossing distance is P-,’= 3.86(O) A. The ionic potential, choosing as zero point of the energy scale the minimum of the ground state, is given by
in
Ui,,( r) = 5352(2.519)
- 11615(0.97)/r,
(13)
are uncertain. The constant term in eq. (13) is where r is in A, Uion(r) in cm-’ and figures in parentheses obtained from ref. [26]; ionization potentiaI(Na) - electron affinity(H) + 0,“. In both cases the crossing distance occurs in a region between v’ = 2 and u’ = 4.
30000
-
25000 ; 5 2
20000
-
15000
-
I 3”
10000
0
I 1
2
3
R(A)
4
5
6
I 7
Fig. 2. The energy gap AU(r) = U,(r) - U,(r) between the A ‘E+ and X ‘Z+ potential curves near the avoided crossing region.
50
A. Pardo et al. / Potential energy curves for the X ‘2 + and A ‘2 + states of NaH
4. Probability density distribution and Franck-Condon
factors
From the eigenfunctions generated in the calculation we have determined distributions 1q”(r) 12 for some vibrational levels. In fig. 3 we present those
*
Fig. 3. Probability density distributions
for the lowest vibrational
R(A)
the probability density functions for the lowest
’
levels (u’ = O-5) of the X ‘2+ and A’Z+
states of NaH.
A. Pardo et al. / Potential energy curoes for the X ‘Z+ and A ‘E+ states of NaH
51
vibrational levels of the X ‘2+ and A ‘Z+ states of NaH. It is observed that the anomalous behavior of the A’Z+ state is clearly shown in the probability density functions of the lowest vibrational levels with a molecular behavior resembling more a harmonic oscillator than an anharmonic one. Even for the u’ = 1 level a changed in asymmetry is observed. These same characteristics have been observed in other hydrides as KH and RbH [27-301. For the X ‘2+ state, sharper probability density distributions than for the A ‘2+ state are obtained showing a situation of greater binding, with a normal behavior in the whole range of vibrational levels. From the obtained wavefunctions we have also determined the Franck-Condon factors for the A ‘2+ @ X ‘Zf system of NaH. In table 10 we show the numerical results obtained through rotationless
Table 10 Franck-Condon
u’
factors (FCF) (for u” = 0,.
,20) for the A’Z+ * X ‘Z+ system of NaH
FCF
0
7.85-6, 0.000117. 0.000782, 0.00322, 0.00935, 0.0209, 0.0381, 0.0586, 0.0788, 0.0947, 0.104, 0.105, 0.099, 0.0883, 0.0748, 0.0607, 0.0474, 0.0357, 0.026, 0.0185, 0.0128
1
0.0000937, 0.00114, 0.00616, 0.0199, 0.0441, 0.0724, 0.0923, 0.0928. 0.0737, 0.0443, 0.0172, 0.00203, 0.00145, 0.0123, 0.0287, 0.0446, 0.0562, 0.0618, 0.0616, 0.057, 0.0497
2
0.00058,0.00566,0.0236,0.0571,0.0888,0.0935,0.0647,0.0242, 0.0000983, 0.00328, 0.0132, 0.0256, 0.0369
3
0.00246, 0.0186, 0.0573, 0.0948, 0.0882, 0.0403, 0.00273, 0.0093, 0.0396, 0.0539, 0.0391, 0.0134, 0.000209, 0.00656, 0.0232, 0.0363, 0.0381, 0.0292, 0.0159, 0.00488, 0.0000975
4
0.00797, 0.0447, 0.0947, 0.0916, 0.0319, 0.0000604, 0.0282, 0.0556, 0.0391, 0.00742, 0.00209, 0.0225, 0.0396, 0.0355, 0.0172, 0.0024, 0.00134, 0.0117, 0.024, 0.0304, 0.0281
5
0.0209, 0.82, 0.105, 0.0397, 0.000405, 0.039, 0.0563, 0.0194, 0.000475, 0.0243, 0.0427, 0.0281, 0.0048, 0.00194, 0.0179, 0.0316, 0.0297, 0.0162, 0.00347, 0.00024, 0.00642
6
0.0453, 0.114, 0.0685, 0.000353, 0.0399, 0.0565, 0.00966, 0.00863, 0.0419, 0.0342, 0.00457, 0.00517, 0.0276, 0.0338, 0.0173, 0.0154, 0.00329, 0.0169, 0.0267, 0.0245, 0.0138
7
0.0829, 0.117, 0.0142, 0.0281, 0.0629, 0.00857, 0.0161, 0.048, 0.0188, 0.000941, 0.0265, 0.03555, 0.0128, 0.0000831, 0.0136, 0.0284, 0.0242, 0.00856, 0.0@00518, 0.00534, 0.0166
8
0.128, 0.077, 0.00468, 0.0707, 0.0169, 0.0164, 0.0508, 0.00995, 0.00972, 0.0394, 0.021, 1.6E-6, 0.0168, 0.0316, 0.0177, 0.00104, 0.00534, 0.0201, 0.0244, 0.0146, 0.00292
9
0.167, 0.0203, 0.0539, 0.0443, 0.00777, 0.0556,0.00833,0.0182, 0.0193, 0.0033, 0.00153, 0.0129, 0.0211
0.0012,0.00657,0.0288,0.0481,0.0519,0.0403,0.0219,0.00657,
0.0416, 0.0069, 0.00876, 0.0335, 0.0196, 0.000215, 0.0116, 0.0268,
10
0.182, 0.00168, 0.087, 0.000369, 0.0583, 0.0148, 0.0206, 0.0419, 0.00112, 0.0235, 0.0327, 0.00303, 0.01, 0.0294, 0.0166, 0.000275, 0.00887, 0.0227, 0.0189, 0.00511, 0.000205
11
0.162, 0.0519, 0.0445,0.0383,0.0376,0.0142,0.047, 0.0129, 0.000129, 0.00762, 0.0195, 0.0175
12
0.114, 0.132, 1.87E-7, 0.079, 0.00131, 0.0582, 0.000624, 0.0407, 0.0167, 0.00881, 0.0354, 0.00676, 0.00781, 0.0283, 0.0127, 0.000235, 0.0148, 0.0221, 0.0091, 2.44E-6, 0.00719
13
0.0606, 0.163, 0.0601, 0.0624, 0.00778, 0.0435, 0.0158, 0.0176, 0.0335, 0.0000238, 0.0253, 0.0215, 4.68E-6, 0.00454, 0.00255, 0.0166, 0.0178, 0.00555
14
0.0218, 0.118, 0.163, 0.0219, 0.0455, 0.0365, 0.0246, 0.0228, 0.0306, 0.00345, 0.035, 0.00597, 0.0109, 0.0269, 0.00506, 0.00489, 0.0212, 0.0135, 0.000335, 0.00648, 0.0167
15
0.00432, 0.046, 0.155, 0.163, 0.0112, 0.0523, 0.021, 0.0333, 0.0106, 0.0352, 0.0000224, 0.0281, 0.0136, 0.0027, 0.0233, 0.012, 0.000242, 0.0137, 0.0174, 0.00442, 0.0085
0.000121, 0.0343, 0.0236, 0.000925, 0.0281, 0.0228, 0.000443, 0.0123, 0.0258,
0.0168, 0.0229,
A. Pardo et al. / Potential energy curves for the X
52
‘2 + and A ‘2 + states of NaH
0.15
kj u.
O.lO_
0.05 -
n
-.-_
“M
1
23
4
5
6
7
8
9
IO
11
12
13
14
t5
Fig. 4. Franck-Condon
16
v’+l
overlap integrals gression.
for the u” = 0 pro-
potentials (J = 0). To explain qualitatively the band intensity distribution in absorption, the Franck-Condon factors (FCF) of the u” = 0 progression are represented in fig. 4. Due to the rapid oscillation of the wavefunctions, a slight shift of some thousandths of A in the potential affects appreciably the intensity distribution. The great number of data for (u’, u”) bands makes their graphic representation difficult, although to provide a qualitative idea of their structures these data have been represented in fig. 5. A typical intensity distribution of the alkali hydrides is observed, where the intensity factors, for low vibrational levels (both u’ and u”), are very small.
NaH A’1
Fig. 5. Threedimensional
graphic
of the Franck-Condon
-x1,.+_
factors
for the A ‘Z + tf X
’ E + band system of NaH.
A. Pardo et al. / Poteniial energy curves for the X
‘2 + and A ‘X
l
slates of NaH
53
Acknowledgement We gratefully acknowledge the support received from the “Comisi6n y TCcnica” (project No. 1203/84) for this research.
Asesora
de Investigacibn
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