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Electronic structure and rovibrational calculation of the low-lying states of the RbYb molecule
Chemical Physics 410 (2013) 37–44
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Electronic structure and rovibrational calculation of the low-lying states of the RbYb molecule S.N. Tohme, M. Korek ⇑ Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Riad El Solh, Beirut 1107 2809, Lebanon
a r t i c l e
i n f o
Article history: Received 11 July 2012 In final form 24 October 2012 Available online 6 November 2012 Keywords: Ab initio calculation Electronic structure Spectroscopic constants Potential energy curves Vibration–rotation calculation
a b s t r a c t Complete Active Space Self Consistent Field (CASSCF) method with Multi Reference Configuration Interaction (MRCI) calculations is used to investigate the potential energy curves of the low-lying 29 electronic states in the representation 2s+1K(+/) of the RbYb molecule (single and double excitations with Davidson corrections). The harmonic frequency xe, the internuclear distance Re and the electronic energy with respect to the ground state Te have been calculated. The eigenvalues Ev, the rotational constant Bv, and the abscissas of the turning points Rmin and Rmax have been investigated using the canonical functions approach. The comparison between the values of the present work and those available in the literature for several states shows a very good agreement. Twenty-six new states have been studied here for the first time. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Advance in trapping, cooling, manipulation and readout of single atoms and ions have led over recent years to a range of fundamental as well as applied investigations on the quantum properties of such systems. Nowadays, an increasing number of experimental groups worldwide are starting experiments with combined charged-neutral systems in various configurations [1]. Theoretical contributions to the field of ultracold molecules are of value in many different respects, but a large number of theoretical calculations are still missing. The determination of accurate molecular potential energy curves (PECs) of ground and relevant electronically excited states is among the most important data. At long-range, atom–atom interactions are typically evaluated by perturbation theory, whereas at short range advanced methods of molecular electronic structure theory come into play [2] where the spectral constants can be extracted (equilibrium bond lengths, harmonic vibrational frequencies, dissociation, and excitation energies) [3– 5]. A large fraction of investigated systems in the ultracold molecular sciences is composed of alkali metal diatomic [6–8]. The RbYb molecule belongs to a new class of heteronuclear diatomics molecules that due to their unpaired electron(s) may be trapped and manipulated using magnetic fields [9–12]. These molecules, for example, are promising candidates for an experimental search for a permanent electric dipole moment of the electron or for producing lattice-spin models [13] for quantum computing. Recently, the ⇑ Corresponding author. Fax: +961 1 818 402. E-mail address: [email protected] (M. Korek). 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2012.10.015
thermalization of various bosonic and fermionic Yb isotopes through collisions with ultracold Rb has been shown, giving first insights into the long-range behavior of the RbYb potential [14]. On the basis of this work, the controlled production of electronically excited RbYb molecules by single-photon photoassociation techniques has been demonstrated [9], and continued efforts include the conservative trapping of the Rb-Yb mixture. Only recently the first ab initio in literature is a complete active space calculation of three doublet electronic states of the RbYb molecule [15]. The splitting among the various J-states is reproduced by Sørensen et al. [16] taking into account the spin-dependent effects. Since the RbYb molecule is an excellent candidate for trapping and for studying dipolar interaction in the ultracold regime, and based on our previous theoretical calculation [17–26], we investigate in the present work more extensive ab initio study of the lowest 29 doublet, and quartet electronic states by using MRDSCI calculations where 26 electronic states have been investigated here for the first time. Taking advantage of the electronic structure of the investigated electronic states of the RbYb molecule and by using the canonical functions approach [27–29], the eigenvalue Ev, the rotational constant Bv, and the abscissas of the turning points Rmin and Rmax have been calculated for several vibrational levels of the considered electronic states.
2. Method of calculations In the present work we study the low-lying doublet and quartet electronic states of the molecule RbYb using state averaged complete active space self consistent field (CASSCF) procedure followed
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S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Fig. 1. Potential energy curves for the lowest doublet R(+/) and D-states of the molecule RbYb.
Fig. 2. Potential energy curves for the lowest doublet P-states of the molecule RbYb.
by a multireference configuration interaction (MRDSCI with Davidson correction) treatment for the electron correlation. The entire CASSCF configuration space was used as the reference in the MRDSCI calculations, which were done via the computational chemistry program MOLPRO [30] taking advantage of the graphical user interface GABEDIT [31]. Rubidium species is treated in all electron scheme; the 37 electrons of the Rubidium atom are considered using a contracted LANL2DZ basis set for s, p functions and Ahlrichs-Cfit for d function. The Ytterbium species is treated as a system of 70 electrons by using the ECP60MHF basis set for s, p, and d functions. Among the 107 electrons explicitly considered for RbYb (37 electrons for Rb and 70 for Yb) 88 inner electrons were frozen in subsequent calculations so that 19 valence electrons were
explicitly treated. All computations were performed in the C2v point group. The potential energy curves of 29 low-lying electronic states of the molecule RbYb were generated using the MRSDCI for 1083 internuclear distances calculations in the range 4.066 Å 6 Re 6 8.626 Å in the representation 2s+1K(+/) where we assumed that, the RbYb molecule is mainly ionic around the equilibrium position.
3. Results The potential energy curves for the 29 low-lying electronic states in the representation 2s+1K(±) have been drawn versus the internuclear distance R in Figs. 1–4. The equilibrium bond dis-
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
39
Fig. 3. Potential energy curves for the lowest quartet R(+/) and D-states of the molecule RbYb.
Fig. 4. Potential energy curves for the lowest quartet P-states of the molecule RbYb.
tances Re, the harmonic vibrational frequencies xe, the relative energy separations Te, and the rotational constants Be, for the doublet and quartet electronic states of RbYb molecule have been obtained by fitting the calculated energy values of the different investigated electronic states into a polynomial in R around the internuclear distance at equilibrium Re (Table 1). The calculated values of the present work in comparison with the results calculated by Sørensen et al. [16] for the ground doublet electronic state shows that, 15 values are in good agreement where the relative difference is 1.3% 6 Dxe/xe 6 8.6%. The different theoretical methods used by Sørensen et al. are the CCSD-SF and CP-CCSD-SF for 3 explicitly cor-
related electrons, the CCSD (T)-SF and the CP-CCSD (T)-SF for 3, 9, and 23 explicitly correlated electrons, and by including the spinor basis to the CCSD-SOC and CCSD (T)-SOC for 9 explicitly correlated electrons [16]. The accuracy for the other 7 values calculated becomes less where Dxe/xe ranges between 10.2% and 17.7%. The internuclear distance Re of the ground state X2R+ of RbYb molecule shows a good agreement in comparison to the 22 values found in literature [16] where the relative difference is 1.5% 6 Dre/ re 6 6.6%. The results of the harmonic frequency xe, and the internuclear distance Re for the (1)2P and (2)2R+ states, were not compared to those obtained theoretically by Sørensen et al. [16] since
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S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Table 1 Spectroscopic constants for the doublet, and quartet electronic states of the molecule RbYb. State 2
X R
+
Te (cm1)
X22 Rþ 1=2
0 0b 0c 0d 0e 0f 0g 0h 0i 0j 0k 0l 0m 0n 0o 0p 0q 0r 0s 0t 0u 0v
X22 Rþ 1=2
0w
2
(1) P
dTe/Te%
a
xe (cm1)
a
27.819 26.278b 26.257c 28.941d 28.923e 24.814f 24.554g 29.724h 29.462i 24.186j 22.882k 28.990l 28.196m 24.073n 28.620o 25.572p 30.214q 24.969r 29.888s 26.443t 31.321u 29.751v
5.5 5.6 4.0 3.9 12.1 11.7 6.8 5.9 13.0 17.7 4.2 1.3 13.4 2.8 8.0 8.6 10.2 7.4 4.9 12.5 6.9
29.458w
5.8
a
P1/2 P3/2 2 P1/2 2 P3/2 (2)2R+ 2 þ R1=2
5029.09 5794 6581 5819 6592 7715.3a 9326
63.584 69.294 69.322 69.441 69.181 44.885a 52.789
(2)2P (1)2P (3)2R+ (1)4R+ (5)2D (3)2P (2)4R (6)2R+ (4)2P (2)4P (2)4R+ (3)4D (3)4P (7)2D (5)2P (8)2R+ (9)2R+ (10)2R+ (4)2D (5)4D (4)4R+ (4)4R (6)4R+ (11)2D
9339 11089.3 11303.4 11558.3 13283.9 13903.3 15244.7 15857.6 16970.0 17070.5 18127.5 19033.4 19909.0 20059.2 20306.8 20658.5 20701.9 21107.4 22896.6 23173.5 23379.9 24091.6 24119.8 25290.2 27721.9
52.284 42.142 43.774 33.859 16.1401 58.795 50.384 64.374 40.932 45.405 39.5107 27.986 166.595 29.025 34.541 39.994 21.443 32.886 29.404 33.582 51.065 37.7908 60.131 32.976 32.506
2
2
dxe/xe%
a
Be 102 (cm1) 1.179
1.661
a
a
1.248a
1.3887 1.357 1.042 0.666 1.6809 1.3888 1.782 1.2347 1.001 1.161 1.168 1.166 0.993 1.1442 1.191 0.7194 0.923 0.638 1.384 1.506 14.469 1.708 0.9805 0.395
Re (Å)
dRe/Re% a
4.99889 4.92134b 4.92134c 4.85784d 4.85784e 4.86843f 4.87901g 4.72555h 4.73084i 4.84197j 4.88430k 4.68851l 4.72555m 4.89325n 4.75042o 4.84035p 4.70281q 4.85093r 4.7081s 4.80332t 4.66578u 4.68165v
1.5 1.5 2.8 2.8 2.6 2.3 5.4 5.3 3.1 2.2 6.2 5.4 2.1 4.9 3.1 5.9 2.9 5.8 3.9 6.6 6.3
4.69752w
6.0
4.2109a 3.9146 3.9304 3.9146 3.9357 4.857a 4.459 4.464 4.604 4.6601 5.314 6.632 4.187 4.606 4.066 4.883 5.403 5.039 5.021 5.254 5.444 5.069 4.968 6.378 5.644 6.795 4.611 4.423 4.517 4.152 5.483 8.626
a
First entry is for the values of the present work. Ref. [16, Table 4]. n?u Ref. [16, Table 5]. w Ref. [16, Table 6]. b?m
their values are obtained by taking into consideration the spin–orbit effect. From this comparison with the available data in literature the accuracy of our calculated values for the new excited electronic states can be confirmed by the experimental investigation in the future for the YbRb molecule. Neither experimental nor theoretical values for the rotational constant Be are available in literature. Table 1 represents for the first time the values of rotational constant Be for the 27 low-lying electronic states. It is quite common, for ground and excited state potential energy curves of molecules, to study a crossing or an
avoided crossing known as conical intersections. These points of the potential energy curves of a diatomic molecule are important in photochemistry. In fact, the avoided crossing regions are likely to be a leakage channels along which the molecules flow from the higher down to the lower potential energy curves. Such crossings or avoided crossings can dramatically alter the stability of molecules. If these crossings are overlooked then low barrier transitions can be missed and an incorrect chemical picture will arise [32]. In the range of R considered, several avoided crossings have been detected in the potential energy curves of the excited elec-
41
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44 Table 2 Positions of the crossings between the different electronic states of the molecule RbYb. State 1
State 2
Crossing between (n1) state1/(n2) state2
Rc (Å)
2
2
3/5 5/4 5/6 7/3 7/8 8/3 8/4 9/4 9/3 3/4 3/10 3/4 4/10 2/1 2/2 2/3 2/3 4/5 4/4 4/6 5/4
3.87 4.85 6.07 3.83 6.23 4.45 4.09 4.13 4.51 5.19 5.47 7.31 5.63 4.39 5.40 5.66 3.77 3.83 4.27 4.99 6.43
R+ 2 D 2
D
2
R+
2
R+
2
R
2
D R
4
4
R+ R
4
4
D
D 2 + R 2 + R 2 R 2 + R 2 R 2 D 2 D 2 R 2 D 2 + R 2 D 2 + R 4 + R 4 + R 4 D 4 D 4 D 4 + R 4 + R 4 + R
3.1. The vibration–rotation calculation By using the canonical functions approach [27–29] and the cubic spline interpolation between each two consecutive points of the potential energy curves obtained from the ab initio calculation of the RbYb molecule, the eigenvalue Ev, the rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning point Rmin and Rmax have been calculated up to the vibrational levels v = 81 for the investigated electronic states. These values for the states X2R+ and (2)2R+ are given in Table 4 and for the states (3)2R+, (5)2P, (1)2D are given in Supplement material file. However, this approach fails if avoided crossings between states occur because of the breakdown of the Born–Oppenheimer approximation at these points. It was not possible to find the values of the eigenvalues Ev, the abscissas of the turning points Rmin and Rmax, the rotational constants Bv, and the centrifugal distortion constants Dv for all the other states due to the existence of either a crossing, or an avoided crossing, or a double minimum. To the best of our knowledge, there are no previous calculations for such parameters, thus, no comparison can be performed.
Table 3 Avoided crossings between different electronic states of the molecule RbYb.
3.2. The permanent dipole moments (PDMs)
State
(n1) state1/(n2) state2
RAC (Å)
DEAC (cm1)
2
8/9 3/4 4/6
5.53 5.65 5.64
283.80 575.78 499.22
R+ P 4 + R 2
tronic states of the molecule RbYb. The positions of these crossings Rc and avoided crossings Rav together with their energy gap separation DEav are reported in Tables 2 and 3, respectively.
The electric dipole moment function Re(r) is given by
Re ðrÞ ¼ hwe jle jw0e i
Table 4 Values of the eigenvalues Ev, the rotational constants Bv, the distortion constant Dv and the abscissas of the turning points for the different vibrational levels of X2R+ and (2)2R+states of the RbYb molecule. X2R+
(2)2R+
v
Ev (cm1)
Bv 103 (cm1)
Dv 109 (cm1)
Rmin (Å)
Rmax (Å)
Ev (cm1)
Bv 103 (cm1)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
14.1 42.1 69.4 96.2 122.5 148.5 173.9 198.9 223.4 247.5 271.2 294.3 317.1 339.4 361.2 382.6 403.5 424.0 444.0 463.6 482.7 501.4 519.6 537.3 554.6 571.5 587.9 603.8 619.3 634.3 648.9 662.9
11.74 11.64 11.55 11.46 11.35 11.25 11.15 11.05 10.94 10.84 10.74 10.63 10.52 10.41 10.31 10.19 10.08 9.97 9.85 9.74 9.62 9.50 9.38 9.26 9.14 9.01 8.88 8.76 8.63 8.50 8.36 8.20
8.19 8.52 8.91 8.67 8.78 9.34 9.32 9.18 9.69 10.05 9.86 9.98 10.57 10.82 10.67 10.86 11.45 11.82 11.78 11.84 12.34 12.95 13.23 13.26 13.45 14.00 14.67 15.17 15.45 15.80 17.02 20.47
4.868 4.774 4.716 4.671 4.634 4.603 4.575 4.550 4.528 4.507 4.489 4.471 4.455 4.440 4.426 4.412 4.400 4.388 4.377 4.366 4.356 4.346 4.337 4.329 4.320 4.312 4.305 4.298 4.291 4.284 4.278 4.272
5.152 5.279 5.374 5.455 5.530 5.600 5.667 5.732 5.795 5.858 5.919 5.980 6.041 6.102 6.162 6.223 6.284 6.346 6.408 6.471 6.534 6.599 6.664 6.730 6.798 6.866 6.936 7.008 7.081 7.156 7.233 7.311
22.1 66.0 109.8 153.4 196.8 240.0 283.1 326.0 368.6 411.2 453.4 495.6 537.5 579.2 620.8 662.1 703.3 744.2 785.0 825.5 865.9 906.0 946.0 985.7 1025.3 1064.6 1103.8 1142.7 1181.4 1219.9 1258.2 1296.3
12.49 12.46 12.43 12.40 12.37 12.34 12.30 12.27 12.24 12.21 12.18 12.14 12.11 12.08 12.04 12.01 11.98 11.94 11.91 11.87 11.84 11.80 11.77 11.73 11.70 11.66 11.62 11.59 11.55 11.51 11.47 11.44
Dv 109 (cm1) 4.03 4.01 4.07 4.05 4.08 4.09 4.09 4.14 4.10 4.19 4.13 4.21 4.18 4.21 4.25 4.22 4.30 4.26 4.32 4.32 4.32 4.39 4.34 4.42 4.42 4.40 4.50 4.45 4.49 4.57 4.49 4.58
Rmin (Å)
Rmax (Å)
4.742 4.664 4.612 4.570 4.535 4.505 4.477 4.452 4.429 4.407 4.387 4.387 4.350 4.333 4.317 4.302 4.287 4.272 4.259 4.246 4.233 4.220 4.208 4.197 4.186 4.175 4.164 4.154 4.144 4.134 4.124 4.115
4.974 5.065 5.130 5.185 5.234 5.278 5.320 5.359 5.396 5.432 5.466 5.499 5.532 5.563 5.594 5.625 5.655 5.684 5.713 5.741 5.770 5.797 5.825 5.852 5.880 5.907 5.933 5.960 5.987 6.013 6.039 6.065
(continued on next page)
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S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Table 4 (continued) X2R+
(2)2R+
v
Ev (cm
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
676.3 689.0 701.6
1
)
3
Bv 10 (cm 8.01 7.86 7.89
1
)
9
Dv 10 (cm 23.62 10.23 9.98
1
)
Rmin (Å)
Rmax (Å)
Ev (cm1)
Bv 103 (cm1)
Dv 109 (cm1)
Rmin (Å)
Rmax (Å)
4.267 4.262 4.257
7.390 7.470 7.554
1334.1 1371.8 1409.2 1446.5 1483.5 1520.3 1556.9 1593.2 1629.4 1665.3 1701.0 1736.5 1771.8 1806.8 1841.6 1876.2 1910.6 1944.7 1978.7 2012.4 2045.8 2079.1 2112.1 2144.9 2177.5 2209.8 2241.9 2273.8 2305.4 2336.8 2368.0 2399.0 2429.7 2460.2 2490.4 2520.4 2550.2 2579.7 2609.0 2638.1 2666.9 2695.5 2723.9 2752.0 2779.8 2807.4 2834.6 2861.1 2886.5 2910.7
11.40 11.36 11.32 11.28 11.24 11.21 11.17 11.13 11.09 11.05 11.01 10.97 10.93 10.88 10.84 10.80 10.76 10.72 10.68 10.63 10.59 10.55 10.50 10.46 10.42 10.37 10.33 10.28 10.24 10.19 10.15 10.10 10.06 10.01 9.97 9.92 9.87 9.82 9.78 9.73 9.68 9.63 9.58 9.53 9.48 9.42 9.34 9.20 8.98 8.84
4.63 4.54 4.67 4.69 4.60 4.75 4.75 4.66 4.81 4.84 4.73 4.87 4.94 4.81 4.91 5.04 4.91 4.94 5.12 5.06 4.98 5.15 5.21 5.08 5.15 5.32 5.25 5.18 5.34 5.43 5.32 5.32 5.50 5.54 5.43 5.48 5.65 5.67 5.58 5.63 5.80 5.83 5.77 5.87 6.30 7.28 9.80 15.45 19.15 1.40
4.106 4.097 4.088 4.080 4.072 4.064 4.056 4.048 4.040 4.033 4.025 4.018 4.011 4.004 3.997 3.991 3.984 3.978 3.971 3.965 3.959 3.953 3.947 3.941 3.936 3.930 3.925 3.919 3.914 3.909 3.903 3.898 3.893 3.888 3.883 3.879 3.874 3.869 3.865 3.860 3.856 3.851 3.847 3.843 3.839 3.835 3.831 3.827 3.823 3.819
6.092 6.118 6.143 6.169 6.195 6.221 6.247 6.273 6.299 6.324 6.350 6.376 6.402 6.428 6.454 6.479 6.505 6.532 6.558 6.584 6.610 6.636 6.663 6.689 6.716 6.742 6.769 6.796 6.823 6.850 6.877 6.905 6.932 6.960 6.988 7.016 7.044 7.072 7.101 7.129 7.158 7.187 7.216 7.246 7.275 7.305 7.335 7.362 7.391 7.408
where we and w0e are respectively the electronic wavefunctions of two different states and le(r) is the permanent dipole moment which can be considered as the response of the wavefunction (and energy) to the external field, in the limit where the field strength is vanishingly small. The dipole moments are analyzed for the 33 lowest electronic states of the RbYb molecule. All the calculations were performed with the MOLPRO [30] program. The dipole moment operator is among the most reliably predicted physical properties, because the quantum mechanical operator is a simple sum of one-electron operators. The expectation value of this operator is sensitive to the nature of the least energetic and most chemically relevant valence electrons [33]. The HF dipole moment is usually large, as the HF wavefunction over estimates the ionic contribution. To obtain the best accuracy, MRCI wavefunctions were constructed using MCSCF active space. The values of the dipole moments for the 2,4R and 2,4D states are given in Debye (D) as a function of the internuclear distance R in Figs. 5 and 6, the
values for the other investigated states are given Supplement material file. The crossing of the potential energy curves and the dipole moment curves for the 2 states (3)2R+ and (5)2D occur at the same value of internuclear distance R = 3.87 Å which is a good sign for the accuracy of the present results. 4. Conclusion In the present work, an ab initio investigation for 29 low-lying electronic states in the representation 2s+1K(+/) of RbYb molecule has been performed via CASSCF/MRCI method (double and quartet excitations). The potential energy curves, the electronic energy with respect to the ground state Te, the harmonic frequency xe, the internuclear distance Re have been calculated along with the rotational constant Be. Taking advantage of the electronic structure of the investigated molecular states of RbYb molecule and by using the canonical function approach, the vibration eigenvalues Ev, the
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
43
Fig. 5. Dipole moment curves of the lowest doublet R(+/) and D-states of the molecule RbYb.
Fig. 6. Dipole moment curves of the lowest quartet R(+/) and D-states of the molecule RbYb.
rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning points Rmin and Rmax were calculated for several vibrational levels for the RbYb molecule. The comparisons of the present results with the available values in the literature show an overall very good agreement. To best of our knowledge, 26 new electronic states have been investigated for the first time through this work. With the recent interest on this molecule through 3 theses [10– 12] during the last 5 years, the present study of these new excited electronic states may lead to more investigation of new experimental works on this molecule.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chemphys.2012. 10.015.
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[20] F. Taher-Mansour, A.R. Allouche, M. Korek, J. Mol. Spectrosc. 248 (2008) 61. [21] M. Korek, S. Kontar, F. Taher-Mansour, A.R. Allouche, Int. J. Quant. Chem. 109 (2009) 236. [22] M. Korek, S. Al-Shawa, G.A. Younes, J. Mol. Struct. THEOCHEM 899 (2009) 25. [23] M. Korek, S.N. Abdul-Al, Chem. Phys. 355 (2009) 130. [24] A. Hamdan, M. Korek, Chem. Phys. 369 (2010) 13. [25] A. Hamdan, M. Korek, Can. J. Chem. 89 (2011) 1304. [26] A. Hamdan, M. Korek, Int. J. Quant. Chem. 111 (2011) 2960. [27] H. Kobeissi, M. Korek, M. Dagher, J. Mol. Spectrosc. 138 (1989) 1. [28] M. Korek, H. Kobeissi, J. Comput. Chem. 13 (1992) 1103. [29] M. Korek, Comput. Phys. Commun. 119 (1999) 169. [30] MOLPRO is a package of ab-intio programs written by H.-J. Werner, P.J. Knowles, R. Lindh, F.R. Manby, M. Schütz, P. Celani, T. Korona, G. Rauhut, R.D. Amos, A. Bernhardsson, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J. Dobbyn, F. Eckert, C. Hampel, G. Hertzer, A.W. Lloyd, S.J. McNicholas, W. Meyer, M.E. Mura, A. Nicklab, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A.J. Stone, R. Tarroni, T. Thorsteinsson. [31] A.R. Allouche, J. Comput. Chem. 32 (2011) 174. [32] D.G. Fedorov, S. Koseki, M.W. Schmidt, M.S. Gordon, Int. Rev. Phys. Chem. 22 (2003) 551. [33] T.C. Steimle, Int. Rev. Phys. Chem. 19 (2000) 455.
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Electronic structure and rovibrational calculation of the low-lying states of the RbYb molecule S.N. Tohme, M. Korek ⇑ Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Riad El Solh, Beirut 1107 2809, Lebanon
a r t i c l e
i n f o
Article history: Received 11 July 2012 In final form 24 October 2012 Available online 6 November 2012 Keywords: Ab initio calculation Electronic structure Spectroscopic constants Potential energy curves Vibration–rotation calculation
a b s t r a c t Complete Active Space Self Consistent Field (CASSCF) method with Multi Reference Configuration Interaction (MRCI) calculations is used to investigate the potential energy curves of the low-lying 29 electronic states in the representation 2s+1K(+/) of the RbYb molecule (single and double excitations with Davidson corrections). The harmonic frequency xe, the internuclear distance Re and the electronic energy with respect to the ground state Te have been calculated. The eigenvalues Ev, the rotational constant Bv, and the abscissas of the turning points Rmin and Rmax have been investigated using the canonical functions approach. The comparison between the values of the present work and those available in the literature for several states shows a very good agreement. Twenty-six new states have been studied here for the first time. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Advance in trapping, cooling, manipulation and readout of single atoms and ions have led over recent years to a range of fundamental as well as applied investigations on the quantum properties of such systems. Nowadays, an increasing number of experimental groups worldwide are starting experiments with combined charged-neutral systems in various configurations [1]. Theoretical contributions to the field of ultracold molecules are of value in many different respects, but a large number of theoretical calculations are still missing. The determination of accurate molecular potential energy curves (PECs) of ground and relevant electronically excited states is among the most important data. At long-range, atom–atom interactions are typically evaluated by perturbation theory, whereas at short range advanced methods of molecular electronic structure theory come into play [2] where the spectral constants can be extracted (equilibrium bond lengths, harmonic vibrational frequencies, dissociation, and excitation energies) [3– 5]. A large fraction of investigated systems in the ultracold molecular sciences is composed of alkali metal diatomic [6–8]. The RbYb molecule belongs to a new class of heteronuclear diatomics molecules that due to their unpaired electron(s) may be trapped and manipulated using magnetic fields [9–12]. These molecules, for example, are promising candidates for an experimental search for a permanent electric dipole moment of the electron or for producing lattice-spin models [13] for quantum computing. Recently, the ⇑ Corresponding author. Fax: +961 1 818 402. E-mail address: [email protected] (M. Korek). 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2012.10.015
thermalization of various bosonic and fermionic Yb isotopes through collisions with ultracold Rb has been shown, giving first insights into the long-range behavior of the RbYb potential [14]. On the basis of this work, the controlled production of electronically excited RbYb molecules by single-photon photoassociation techniques has been demonstrated [9], and continued efforts include the conservative trapping of the Rb-Yb mixture. Only recently the first ab initio in literature is a complete active space calculation of three doublet electronic states of the RbYb molecule [15]. The splitting among the various J-states is reproduced by Sørensen et al. [16] taking into account the spin-dependent effects. Since the RbYb molecule is an excellent candidate for trapping and for studying dipolar interaction in the ultracold regime, and based on our previous theoretical calculation [17–26], we investigate in the present work more extensive ab initio study of the lowest 29 doublet, and quartet electronic states by using MRDSCI calculations where 26 electronic states have been investigated here for the first time. Taking advantage of the electronic structure of the investigated electronic states of the RbYb molecule and by using the canonical functions approach [27–29], the eigenvalue Ev, the rotational constant Bv, and the abscissas of the turning points Rmin and Rmax have been calculated for several vibrational levels of the considered electronic states.
2. Method of calculations In the present work we study the low-lying doublet and quartet electronic states of the molecule RbYb using state averaged complete active space self consistent field (CASSCF) procedure followed
38
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Fig. 1. Potential energy curves for the lowest doublet R(+/) and D-states of the molecule RbYb.
Fig. 2. Potential energy curves for the lowest doublet P-states of the molecule RbYb.
by a multireference configuration interaction (MRDSCI with Davidson correction) treatment for the electron correlation. The entire CASSCF configuration space was used as the reference in the MRDSCI calculations, which were done via the computational chemistry program MOLPRO [30] taking advantage of the graphical user interface GABEDIT [31]. Rubidium species is treated in all electron scheme; the 37 electrons of the Rubidium atom are considered using a contracted LANL2DZ basis set for s, p functions and Ahlrichs-Cfit for d function. The Ytterbium species is treated as a system of 70 electrons by using the ECP60MHF basis set for s, p, and d functions. Among the 107 electrons explicitly considered for RbYb (37 electrons for Rb and 70 for Yb) 88 inner electrons were frozen in subsequent calculations so that 19 valence electrons were
explicitly treated. All computations were performed in the C2v point group. The potential energy curves of 29 low-lying electronic states of the molecule RbYb were generated using the MRSDCI for 1083 internuclear distances calculations in the range 4.066 Å 6 Re 6 8.626 Å in the representation 2s+1K(+/) where we assumed that, the RbYb molecule is mainly ionic around the equilibrium position.
3. Results The potential energy curves for the 29 low-lying electronic states in the representation 2s+1K(±) have been drawn versus the internuclear distance R in Figs. 1–4. The equilibrium bond dis-
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
39
Fig. 3. Potential energy curves for the lowest quartet R(+/) and D-states of the molecule RbYb.
Fig. 4. Potential energy curves for the lowest quartet P-states of the molecule RbYb.
tances Re, the harmonic vibrational frequencies xe, the relative energy separations Te, and the rotational constants Be, for the doublet and quartet electronic states of RbYb molecule have been obtained by fitting the calculated energy values of the different investigated electronic states into a polynomial in R around the internuclear distance at equilibrium Re (Table 1). The calculated values of the present work in comparison with the results calculated by Sørensen et al. [16] for the ground doublet electronic state shows that, 15 values are in good agreement where the relative difference is 1.3% 6 Dxe/xe 6 8.6%. The different theoretical methods used by Sørensen et al. are the CCSD-SF and CP-CCSD-SF for 3 explicitly cor-
related electrons, the CCSD (T)-SF and the CP-CCSD (T)-SF for 3, 9, and 23 explicitly correlated electrons, and by including the spinor basis to the CCSD-SOC and CCSD (T)-SOC for 9 explicitly correlated electrons [16]. The accuracy for the other 7 values calculated becomes less where Dxe/xe ranges between 10.2% and 17.7%. The internuclear distance Re of the ground state X2R+ of RbYb molecule shows a good agreement in comparison to the 22 values found in literature [16] where the relative difference is 1.5% 6 Dre/ re 6 6.6%. The results of the harmonic frequency xe, and the internuclear distance Re for the (1)2P and (2)2R+ states, were not compared to those obtained theoretically by Sørensen et al. [16] since
40
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Table 1 Spectroscopic constants for the doublet, and quartet electronic states of the molecule RbYb. State 2
X R
+
Te (cm1)
X22 Rþ 1=2
0 0b 0c 0d 0e 0f 0g 0h 0i 0j 0k 0l 0m 0n 0o 0p 0q 0r 0s 0t 0u 0v
X22 Rþ 1=2
0w
2
(1) P
dTe/Te%
a
xe (cm1)
a
27.819 26.278b 26.257c 28.941d 28.923e 24.814f 24.554g 29.724h 29.462i 24.186j 22.882k 28.990l 28.196m 24.073n 28.620o 25.572p 30.214q 24.969r 29.888s 26.443t 31.321u 29.751v
5.5 5.6 4.0 3.9 12.1 11.7 6.8 5.9 13.0 17.7 4.2 1.3 13.4 2.8 8.0 8.6 10.2 7.4 4.9 12.5 6.9
29.458w
5.8
a
P1/2 P3/2 2 P1/2 2 P3/2 (2)2R+ 2 þ R1=2
5029.09 5794 6581 5819 6592 7715.3a 9326
63.584 69.294 69.322 69.441 69.181 44.885a 52.789
(2)2P (1)2P (3)2R+ (1)4R+ (5)2D (3)2P (2)4R (6)2R+ (4)2P (2)4P (2)4R+ (3)4D (3)4P (7)2D (5)2P (8)2R+ (9)2R+ (10)2R+ (4)2D (5)4D (4)4R+ (4)4R (6)4R+ (11)2D
9339 11089.3 11303.4 11558.3 13283.9 13903.3 15244.7 15857.6 16970.0 17070.5 18127.5 19033.4 19909.0 20059.2 20306.8 20658.5 20701.9 21107.4 22896.6 23173.5 23379.9 24091.6 24119.8 25290.2 27721.9
52.284 42.142 43.774 33.859 16.1401 58.795 50.384 64.374 40.932 45.405 39.5107 27.986 166.595 29.025 34.541 39.994 21.443 32.886 29.404 33.582 51.065 37.7908 60.131 32.976 32.506
2
2
dxe/xe%
a
Be 102 (cm1) 1.179
1.661
a
a
1.248a
1.3887 1.357 1.042 0.666 1.6809 1.3888 1.782 1.2347 1.001 1.161 1.168 1.166 0.993 1.1442 1.191 0.7194 0.923 0.638 1.384 1.506 14.469 1.708 0.9805 0.395
Re (Å)
dRe/Re% a
4.99889 4.92134b 4.92134c 4.85784d 4.85784e 4.86843f 4.87901g 4.72555h 4.73084i 4.84197j 4.88430k 4.68851l 4.72555m 4.89325n 4.75042o 4.84035p 4.70281q 4.85093r 4.7081s 4.80332t 4.66578u 4.68165v
1.5 1.5 2.8 2.8 2.6 2.3 5.4 5.3 3.1 2.2 6.2 5.4 2.1 4.9 3.1 5.9 2.9 5.8 3.9 6.6 6.3
4.69752w
6.0
4.2109a 3.9146 3.9304 3.9146 3.9357 4.857a 4.459 4.464 4.604 4.6601 5.314 6.632 4.187 4.606 4.066 4.883 5.403 5.039 5.021 5.254 5.444 5.069 4.968 6.378 5.644 6.795 4.611 4.423 4.517 4.152 5.483 8.626
a
First entry is for the values of the present work. Ref. [16, Table 4]. n?u Ref. [16, Table 5]. w Ref. [16, Table 6]. b?m
their values are obtained by taking into consideration the spin–orbit effect. From this comparison with the available data in literature the accuracy of our calculated values for the new excited electronic states can be confirmed by the experimental investigation in the future for the YbRb molecule. Neither experimental nor theoretical values for the rotational constant Be are available in literature. Table 1 represents for the first time the values of rotational constant Be for the 27 low-lying electronic states. It is quite common, for ground and excited state potential energy curves of molecules, to study a crossing or an
avoided crossing known as conical intersections. These points of the potential energy curves of a diatomic molecule are important in photochemistry. In fact, the avoided crossing regions are likely to be a leakage channels along which the molecules flow from the higher down to the lower potential energy curves. Such crossings or avoided crossings can dramatically alter the stability of molecules. If these crossings are overlooked then low barrier transitions can be missed and an incorrect chemical picture will arise [32]. In the range of R considered, several avoided crossings have been detected in the potential energy curves of the excited elec-
41
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44 Table 2 Positions of the crossings between the different electronic states of the molecule RbYb. State 1
State 2
Crossing between (n1) state1/(n2) state2
Rc (Å)
2
2
3/5 5/4 5/6 7/3 7/8 8/3 8/4 9/4 9/3 3/4 3/10 3/4 4/10 2/1 2/2 2/3 2/3 4/5 4/4 4/6 5/4
3.87 4.85 6.07 3.83 6.23 4.45 4.09 4.13 4.51 5.19 5.47 7.31 5.63 4.39 5.40 5.66 3.77 3.83 4.27 4.99 6.43
R+ 2 D 2
D
2
R+
2
R+
2
R
2
D R
4
4
R+ R
4
4
D
D 2 + R 2 + R 2 R 2 + R 2 R 2 D 2 D 2 R 2 D 2 + R 2 D 2 + R 4 + R 4 + R 4 D 4 D 4 D 4 + R 4 + R 4 + R
3.1. The vibration–rotation calculation By using the canonical functions approach [27–29] and the cubic spline interpolation between each two consecutive points of the potential energy curves obtained from the ab initio calculation of the RbYb molecule, the eigenvalue Ev, the rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning point Rmin and Rmax have been calculated up to the vibrational levels v = 81 for the investigated electronic states. These values for the states X2R+ and (2)2R+ are given in Table 4 and for the states (3)2R+, (5)2P, (1)2D are given in Supplement material file. However, this approach fails if avoided crossings between states occur because of the breakdown of the Born–Oppenheimer approximation at these points. It was not possible to find the values of the eigenvalues Ev, the abscissas of the turning points Rmin and Rmax, the rotational constants Bv, and the centrifugal distortion constants Dv for all the other states due to the existence of either a crossing, or an avoided crossing, or a double minimum. To the best of our knowledge, there are no previous calculations for such parameters, thus, no comparison can be performed.
Table 3 Avoided crossings between different electronic states of the molecule RbYb.
3.2. The permanent dipole moments (PDMs)
State
(n1) state1/(n2) state2
RAC (Å)
DEAC (cm1)
2
8/9 3/4 4/6
5.53 5.65 5.64
283.80 575.78 499.22
R+ P 4 + R 2
tronic states of the molecule RbYb. The positions of these crossings Rc and avoided crossings Rav together with their energy gap separation DEav are reported in Tables 2 and 3, respectively.
The electric dipole moment function Re(r) is given by
Re ðrÞ ¼ hwe jle jw0e i
Table 4 Values of the eigenvalues Ev, the rotational constants Bv, the distortion constant Dv and the abscissas of the turning points for the different vibrational levels of X2R+ and (2)2R+states of the RbYb molecule. X2R+
(2)2R+
v
Ev (cm1)
Bv 103 (cm1)
Dv 109 (cm1)
Rmin (Å)
Rmax (Å)
Ev (cm1)
Bv 103 (cm1)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
14.1 42.1 69.4 96.2 122.5 148.5 173.9 198.9 223.4 247.5 271.2 294.3 317.1 339.4 361.2 382.6 403.5 424.0 444.0 463.6 482.7 501.4 519.6 537.3 554.6 571.5 587.9 603.8 619.3 634.3 648.9 662.9
11.74 11.64 11.55 11.46 11.35 11.25 11.15 11.05 10.94 10.84 10.74 10.63 10.52 10.41 10.31 10.19 10.08 9.97 9.85 9.74 9.62 9.50 9.38 9.26 9.14 9.01 8.88 8.76 8.63 8.50 8.36 8.20
8.19 8.52 8.91 8.67 8.78 9.34 9.32 9.18 9.69 10.05 9.86 9.98 10.57 10.82 10.67 10.86 11.45 11.82 11.78 11.84 12.34 12.95 13.23 13.26 13.45 14.00 14.67 15.17 15.45 15.80 17.02 20.47
4.868 4.774 4.716 4.671 4.634 4.603 4.575 4.550 4.528 4.507 4.489 4.471 4.455 4.440 4.426 4.412 4.400 4.388 4.377 4.366 4.356 4.346 4.337 4.329 4.320 4.312 4.305 4.298 4.291 4.284 4.278 4.272
5.152 5.279 5.374 5.455 5.530 5.600 5.667 5.732 5.795 5.858 5.919 5.980 6.041 6.102 6.162 6.223 6.284 6.346 6.408 6.471 6.534 6.599 6.664 6.730 6.798 6.866 6.936 7.008 7.081 7.156 7.233 7.311
22.1 66.0 109.8 153.4 196.8 240.0 283.1 326.0 368.6 411.2 453.4 495.6 537.5 579.2 620.8 662.1 703.3 744.2 785.0 825.5 865.9 906.0 946.0 985.7 1025.3 1064.6 1103.8 1142.7 1181.4 1219.9 1258.2 1296.3
12.49 12.46 12.43 12.40 12.37 12.34 12.30 12.27 12.24 12.21 12.18 12.14 12.11 12.08 12.04 12.01 11.98 11.94 11.91 11.87 11.84 11.80 11.77 11.73 11.70 11.66 11.62 11.59 11.55 11.51 11.47 11.44
Dv 109 (cm1) 4.03 4.01 4.07 4.05 4.08 4.09 4.09 4.14 4.10 4.19 4.13 4.21 4.18 4.21 4.25 4.22 4.30 4.26 4.32 4.32 4.32 4.39 4.34 4.42 4.42 4.40 4.50 4.45 4.49 4.57 4.49 4.58
Rmin (Å)
Rmax (Å)
4.742 4.664 4.612 4.570 4.535 4.505 4.477 4.452 4.429 4.407 4.387 4.387 4.350 4.333 4.317 4.302 4.287 4.272 4.259 4.246 4.233 4.220 4.208 4.197 4.186 4.175 4.164 4.154 4.144 4.134 4.124 4.115
4.974 5.065 5.130 5.185 5.234 5.278 5.320 5.359 5.396 5.432 5.466 5.499 5.532 5.563 5.594 5.625 5.655 5.684 5.713 5.741 5.770 5.797 5.825 5.852 5.880 5.907 5.933 5.960 5.987 6.013 6.039 6.065
(continued on next page)
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S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
Table 4 (continued) X2R+
(2)2R+
v
Ev (cm
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
676.3 689.0 701.6
1
)
3
Bv 10 (cm 8.01 7.86 7.89
1
)
9
Dv 10 (cm 23.62 10.23 9.98
1
)
Rmin (Å)
Rmax (Å)
Ev (cm1)
Bv 103 (cm1)
Dv 109 (cm1)
Rmin (Å)
Rmax (Å)
4.267 4.262 4.257
7.390 7.470 7.554
1334.1 1371.8 1409.2 1446.5 1483.5 1520.3 1556.9 1593.2 1629.4 1665.3 1701.0 1736.5 1771.8 1806.8 1841.6 1876.2 1910.6 1944.7 1978.7 2012.4 2045.8 2079.1 2112.1 2144.9 2177.5 2209.8 2241.9 2273.8 2305.4 2336.8 2368.0 2399.0 2429.7 2460.2 2490.4 2520.4 2550.2 2579.7 2609.0 2638.1 2666.9 2695.5 2723.9 2752.0 2779.8 2807.4 2834.6 2861.1 2886.5 2910.7
11.40 11.36 11.32 11.28 11.24 11.21 11.17 11.13 11.09 11.05 11.01 10.97 10.93 10.88 10.84 10.80 10.76 10.72 10.68 10.63 10.59 10.55 10.50 10.46 10.42 10.37 10.33 10.28 10.24 10.19 10.15 10.10 10.06 10.01 9.97 9.92 9.87 9.82 9.78 9.73 9.68 9.63 9.58 9.53 9.48 9.42 9.34 9.20 8.98 8.84
4.63 4.54 4.67 4.69 4.60 4.75 4.75 4.66 4.81 4.84 4.73 4.87 4.94 4.81 4.91 5.04 4.91 4.94 5.12 5.06 4.98 5.15 5.21 5.08 5.15 5.32 5.25 5.18 5.34 5.43 5.32 5.32 5.50 5.54 5.43 5.48 5.65 5.67 5.58 5.63 5.80 5.83 5.77 5.87 6.30 7.28 9.80 15.45 19.15 1.40
4.106 4.097 4.088 4.080 4.072 4.064 4.056 4.048 4.040 4.033 4.025 4.018 4.011 4.004 3.997 3.991 3.984 3.978 3.971 3.965 3.959 3.953 3.947 3.941 3.936 3.930 3.925 3.919 3.914 3.909 3.903 3.898 3.893 3.888 3.883 3.879 3.874 3.869 3.865 3.860 3.856 3.851 3.847 3.843 3.839 3.835 3.831 3.827 3.823 3.819
6.092 6.118 6.143 6.169 6.195 6.221 6.247 6.273 6.299 6.324 6.350 6.376 6.402 6.428 6.454 6.479 6.505 6.532 6.558 6.584 6.610 6.636 6.663 6.689 6.716 6.742 6.769 6.796 6.823 6.850 6.877 6.905 6.932 6.960 6.988 7.016 7.044 7.072 7.101 7.129 7.158 7.187 7.216 7.246 7.275 7.305 7.335 7.362 7.391 7.408
where we and w0e are respectively the electronic wavefunctions of two different states and le(r) is the permanent dipole moment which can be considered as the response of the wavefunction (and energy) to the external field, in the limit where the field strength is vanishingly small. The dipole moments are analyzed for the 33 lowest electronic states of the RbYb molecule. All the calculations were performed with the MOLPRO [30] program. The dipole moment operator is among the most reliably predicted physical properties, because the quantum mechanical operator is a simple sum of one-electron operators. The expectation value of this operator is sensitive to the nature of the least energetic and most chemically relevant valence electrons [33]. The HF dipole moment is usually large, as the HF wavefunction over estimates the ionic contribution. To obtain the best accuracy, MRCI wavefunctions were constructed using MCSCF active space. The values of the dipole moments for the 2,4R and 2,4D states are given in Debye (D) as a function of the internuclear distance R in Figs. 5 and 6, the
values for the other investigated states are given Supplement material file. The crossing of the potential energy curves and the dipole moment curves for the 2 states (3)2R+ and (5)2D occur at the same value of internuclear distance R = 3.87 Å which is a good sign for the accuracy of the present results. 4. Conclusion In the present work, an ab initio investigation for 29 low-lying electronic states in the representation 2s+1K(+/) of RbYb molecule has been performed via CASSCF/MRCI method (double and quartet excitations). The potential energy curves, the electronic energy with respect to the ground state Te, the harmonic frequency xe, the internuclear distance Re have been calculated along with the rotational constant Be. Taking advantage of the electronic structure of the investigated molecular states of RbYb molecule and by using the canonical function approach, the vibration eigenvalues Ev, the
S.N. Tohme, M. Korek / Chemical Physics 410 (2013) 37–44
43
Fig. 5. Dipole moment curves of the lowest doublet R(+/) and D-states of the molecule RbYb.
Fig. 6. Dipole moment curves of the lowest quartet R(+/) and D-states of the molecule RbYb.
rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning points Rmin and Rmax were calculated for several vibrational levels for the RbYb molecule. The comparisons of the present results with the available values in the literature show an overall very good agreement. To best of our knowledge, 26 new electronic states have been investigated for the first time through this work. With the recent interest on this molecule through 3 theses [10– 12] during the last 5 years, the present study of these new excited electronic states may lead to more investigation of new experimental works on this molecule.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chemphys.2012. 10.015.
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