- Email: [email protected]
Line list for the ground state of CaF
Accepted Manuscript
Line list for the ground state of CaF Shilin Hou , Peter F. Bernath PII: DOI: Reference:
S0022-4073(17)30980-9 10.1016/j.jqsrt.2018.02.011 JQSRT 5992
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
29 December 2017 7 February 2018 7 February 2018
Please cite this article as: Shilin Hou , Peter F. Bernath , Line list for the ground state of CaF, Journal of Quantitative Spectroscopy & Radiative Transfer (2018), doi: 10.1016/j.jqsrt.2018.02.011
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Highlights Analytical Extended Morse Oscillator (EMO) potential function derived for CaF
Ground state dipole moment function calculated ab initio
Line lists calculated for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF vibration-rotation transitions
Useful for spectra of high temperature environments such as hot super-Earth exoplanets
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Line list for the ground state of CaF Shilin Hou a and Peter F. Bernath b,* a b
Department of Physics, Ocean University of China, Qingdao, Shandong 266100, China
Department of Chemistry and Biochemistry, Old Dominion University, Norfolk, Virginia 23529, USA
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*Corresponding author. E-mail address: [email protected] (P.F. Bernath).
Abstract
The molecular potential energy function and electronic dipole moment function for the ground state of
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CaF were studied with MRCI, ACPF, and RCCSD(T) ab initio calculations. The RCCSD(T) potential function reproduces the experimental vibrational intervals to within ~2 cm-1. The RCCSD(T) dipole moment at the equilibrium internuclear separation agrees well with the experimental value. Over a wide range of internuclear separations, far beyond the range associated with the observed spectra, the ab initio dipole moment functions are similar and highly linear. An extended Morse oscillator (EMO) potential function was also obtained by fitting the observed lines of the laboratory vibration-rotation and pure
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rotation spectra of the 40CaF X2Σ+ ground state. The fitted potential reproduces the observed transitions (v≤8, N≤121, Δv=0, 1) within their experimental uncertainties. With this EMO potential and the
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RCCSD(T) dipole moment function, line lists for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF, and 48CaF were computed for v≤10, N≤121, Δv=0–10. The calculated emission spectra are in good agreement with an
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observed laboratory spectrum of CaF at a sample temperature of 1873 K.
Keywords: Molecular potential; dipole moment function; Einstein A coefficients; vibration-rotation line
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1 Introduction Calcium fluoride is a molecule of interest in many fields such as astrophysics, spectroscopy, computational chemistry, and in cold molecule trapping [e.g., 1-21]. Optical, infrared, and microwave spectra have been obtained and analyzed for transitions associated with the ground state and many excited electronic states [2-5, 9-13]. For example, electronic transitions involving the A2Π，B2Σ+, C2Π,
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E2Σ+ and E'2Π states [2-5], and many Rydberg states [18, 20] are known. Recently the CaF B-X system was used to improve the analytical capabilities of LIBS (laser-induced breakdown spectroscopy) fluorine detection in Ca-containing samples as well as in calcium-free samples [22].
For the ground state, vibration-rotation lines for v=0–8 and N up to 121 were analyzed by Charron et al. [11] from an infrared emission spectrum; millimeter-wave transitions were observed by Anderson et al.
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[9] for v=0–3, N from 3 to 18. Stark splittings were studied by Childs et al. [6,7] for v=0-1, from which the dipole moments for v=0 (3.07(7) debye; D) and v=1 (3.12(7) D) were obtained; the dipole moment at the equilibrium internuclear separation was derived as 3.05(7) D.
In circumstellar environments, such around the cool carbon star IRC+10216, metal salts containing
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sodium and aluminum, such as AlCl, AlF, and NaCl, have already been detected [23]. Because the cosmic abundance of calcium is similar to that of aluminum and sodium, other metal fluorides such as CaF may
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be detectable also [10].
As a free radical with a single electron outside closed-shell ionic cores, CaF is also of special theoretical
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significance [8,12,14-20] especially for studies of the properties of its Rydberg states [12,18,20]. For the ground state, highly ionic bonding (Ca+F-) near equilibrium is well-known and is supported by the
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relatively small value of the hyperfine constant b, which includes the Fermi contact electron spin density term [1]. This implies that the electron density at the F nucleus (which has the nuclear spin) is not very
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large as expected if the lone unpaired electron is primarily located on the metal atom.
The purpose of the present work is to prepare a CaF vibration-rotation line list with positions and line strengths. In order to obtain line strengths, a dipole moment function is computed with high quality ab initio methods. The CaF line list can be used for quantitative analysis and prediction of infrared spectra of hot CaF as found for example in hot super-Earth exoplanets. The ro-vibrational line positions (and line intensities) are obtained by direct solution of the Schrödinger
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equation for nuclear motion using Le Roy's computer program LEVEL [24] with an accurate empirical potential energy function. The potential energy function was obtained by fitting the pure rotation [9] and vibration-rotation [11] data using LeRoy's dPotFit code [25]. The potential function used was an Expanded Morse Oscillator (EMO) [26] similar to the work of Barton et al. [27] on NaCl and KCl.
The ground state potential energy function was studied with both ab initio and empirical approaches. The
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ab initio data were obtained to study the behavior over the full range of the potential; the empirical
potential was used to improve the accuracy of the predicted vibration-rotation transitions. All ab initio computations were performed with the Molpro [28] package.
The dipole moment at the equilibrium internuclear separation, re, which can be derived from the
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experimental data for the v=0 and 1 levels, was used as an important criterion to choose the best dipole moment function from the ab initio computations. The calculated line intensities are very sensitive to the form of the dipole moment function and its derivatives [29]; therefore, dipole moment functions from several different ab initio approaches were compared, and the best one was then used in the LEVEL program to calculate the line intensities for 40CaF and its isotopologues. This calculation also gives a high
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quality molecular potential function which best reproduces the observed spectroscopic data.
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2 Fitted molecular potential from observed spectra The line positions of CaF can be reproduced using Le Roy’s extended Morse oscillator (EMO) potential
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model [26], which can be expressed as
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where
( )(
( )
( )
∑
(
)
)
(
(1)
) .
(2) (3)
De is the equilibrium potential well depth and re is the equilibrium internuclear distance. N and p are preset parameters, and the i parameters are determined using Le Roy’s potential fitting program dPotFit [25]. p was set to 3, N was set to 5, De was fixed and re was set free in the fit. Information for the observed transitions adopted for fitting is listed in Table 1.
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De was set to 44203.50 cm−1, which was obtained using De=D0+E0, with D0 taken from Haynes [30], and the zero-point energy E0 calculated from the spectroscopic constants of Charron et al. [11]. In total 811 observed transitions were fitted with a dimensionless standard error (DSE) of 0.89. The 40CaF pure rotation data were corrected for the effects of fine and hyperfine structure as in Charron et al. [11]. For the vibration-rotation lines, the uncertainties assigned in the original experimental work [11], in the range of 0.001 cm−1 − 0.005 cm−1, were used and for the millimeter-wave data [9], an uncertainty of 9.3×10−7
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cm−1 was adopted. These uncertainties correspond to one standard deviation. The small spin splitting and F hyperfine structure was not resolved in the vibration-rotation spectrum [ 11] recorded at 0.01 cm−1 resolution and CaF was treated as if the ground state has 1Σ+ symmetry. In the present work, we aim to provide a vibration-rotation line list so we follow Charron et al. [11] and ignore fine and hyperfine
structure. The EMO potential parameters obtained from dPotFit are presented in Table 2 along with 95%
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confidence limits.
Table 1. Laboratory data used to determine the CaF potential energy curve. Transitions CaFJ ∆N N
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Charron, et al. 1995 [11] T=1873 K
∆N
Uncertainty (cm-1)
1.7x10-5 cm-1
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Table 2. EMO potential parameters for CaF. CaF EMO(N=5, p=3) De/cm-1 44203.50 re/Å 1.951636551397 1.223498191939D+00 -2.618924324279D-01 -6.058869340934D-02 5.708544137802D-02 -5.248856409990D-02 7.362231681793D-02
Uncertainty Fixed 2.5D-07 1.0D-06 2.2D-05 9.3D-05 8.2D-04 2.7D-03 1.1D-02
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0.001-0.005 cm-1
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Anderson, et al. 1994 [9]
Frequency range (cm-1)
3 Ab initio dipole moment functions and molecular potentials To obtain a reliable dipole moment function for calculating transition dipole moments, several high-level ab initio calculations were performed with various large basis sets. Methods used include the internallycontracted multiconfiguration reference configuration interaction (MRCI) [31-33], the averaged coupledpair functional approach (ACPF) [34-35], and the partially spin restricted open-shell coupled cluster theory with single and double excitations plus a correction for triple excitations (RCCSD(T)) [36].
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Correlation-consistent basis sets [37-41] were used in the calculations. Basis functions mainly include ccpvxz, cc-pcvxz for Ca and/or F, and aug-cc-pvxz for F with x=Q and 5 [37-41]. All calculations were performed with Molpro, using the basis sets embedded in the package.
For the ground state of CaF, the 29 electrons are distributed in 15 molecular orbitals, with orbital energy
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order at the equilibrium geometry 1σ22σ23σ24σ21π45σ26σ22π47σ23π48σ29σ1. The electronic configuration can be expressed as (9a1, 3b1, 3b2, 0a2) (more compactly (9,3,3,0) or (9,3,3)) using the irreducible
representations of the C2v point group used by Molpro. The wavefunctions from state-averaged complete active space self-consistent field (CASSCF) [42,43] calculations were used as reference wavefunctions for the MRCI and ACPF calculations. In most of the MRCI and ACPF calculations, the active space, the doubly occupied space, and frozen/core space are the same as that of the corresponding CASSCF
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calculations. When the occupied orbitals were set to be (9,3,3) and (10,4,4) (in which the s and p atomic orbitals were included in the reference), the doubly occupied orbitals were set to be (6,2,2) or (4,1,1) and the frozen orbitals were set to be the same as the doubly occupied space or smaller, the obtained potential curves and the dipole moment functions were smooth, but the computed dipole moment functions were not converged at large distances. The ACPF and MRCI dipole moments obtained were about ~10% lower
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than the observed value at the equilibrium geometry.
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MgF (dipole moment ~3.004 D) [44] and CaF (~3.05 D) have similar dipole moments at equilibrium; therefore, some similarity in the dipole moment function is expected. To obtain a converged dipole moment function at large distances, the active space was enlarged, and d atomic orbitals (of Ca) were
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included in the active space, i.e., the occupied orbitals were set to be (12,5,5,1). The doubly occupied space was (6,2,2); thus, the active space was set as {6,3,3,1} for the remaining 9 valence electrons in the
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corresponding CASSCF/ACPF (or MRCI) calculations. Some typical results are shown in Figures 1 and 2
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for comparison.
When the doubly occupied, frozen core space was (6,2,2), converged dipole moment functions were obtained; but the calculated potential curves had a second minimum at about 2.5-3.0 Å (shown with “+” and “△” symbols in Figure 1), which is not physically correct. To solve this problem, the frozen/core spaces in MRCI, ACPF, and CASSCF computations were set to be
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(4,1,1). The cc-pvqz basis sets were tried for both Ca and F. Note that the cost for such a calculation is large at present. For example, in the MRCI/cc-pvqz calculation, there are in total 107,694 configuration state functions (CSFs) involved in the reference space, and 64,990,638 internal contracted CSFs involved in the MRCI configuration space. After more than one month of
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computation on a supercomputer, only 13 points in the range of 1.4-3.9 Å were obtained (shown with “*” symbols in Figure 1). As expected, the second minimum disappeared, and both the potential curve and the dipole moment function are smooth as well. To get a more economical solution, we tried a mixed scheme, i.e., in the CASSCF calculation, the frozen space was set to
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be (4,1,1), while in the following ACPF (or MRCI) calculations, the core spaces were set to be (6,2,2). The calculated ACPF and MRCI potential are smooth, and the dipole moment functions are similar to that obtained with the frozen/core spaces both set to (4,1,1). Considerable
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computation time was saved in this way. However, the MRCI and ACPF dipole moments were
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not improved near equilibrium compared to the observed data [6].
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The comparison between dipole moments from difference methods is given in Table 3. In this table, literature results with CPF (coupled pair functional) [8] and RCCSD(T) [16] approaches gave the predictions of 3.06 D [8] and 3.05 D [16], respectively, which are nearly identical to our RCCSD(T)
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prediction (3.04 D). These predictions are close to the experimental equilibrium dipole moment of 3.05(7) D given by Childs et al. [7]. Most of the other predictions in Table 3 differ by relatively large amounts
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(>10%) from this value.
In our RCCSD(T) calculations, different basis set were also tried; finally, we found that the RCCSD(T) approach with basis sets for Ca=cc-pv5z and F=aug-cc-pv5z gave the best dipole as well as molecular potential in the range of observed data. The dipole moment points were
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obtained with the finite field method embedded in Molpro, with the electric field set to the small value of 0.0005 a.u. (see for example, Ref. [45]). The obtained dipole moment at the optimized re (1.9526 Å) is 3.04 D (1.1947 a.u.), which agrees well with the observed value 3.05(7) D [7]. This RCCSD(T) re value also agrees well with the experimental value of re=1.9516 Å [11]. When the
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potential well is shifted to the EMO potential, the RCCSD(T) potential agrees very well with the EMO potential. A comparison between the EMO potential and the RCCSD(T) data is shown in Figure 3. The calculated ∆Gv values from this RCCSD(T) potential agree well with the experimental data (differences ~2 cm-1), as presented in Table 4. The largest relative
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discrepancy is less than 0.4%. The RCCSD(T) dipole moment function is plotted in Figure 2.
The results in Figure 2 show that some of the dipole moment functions do not converge. The
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dipole moment functions are very linear up to about 6 Å (except for ACPF/cc-pcvqz) for MRCI/ACPF with relatively large basis sets, which is far large than the range (less than 3 Å)
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sampled by the observed transitions of Charron et al. [11]. Since the RCCSD(T) approach gives
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the best value compared to the observed value at equilibrium, and converged to the high-level
ab inito results (e.g., ACPF/(Ca:v5z;F:av5z) and MRCI/cc-pcv5z in Figure 2 ) at large distances,
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the RCCSD(T) dipole moment function was used for the line intensity calculation. A comparison between dipole moment functions around equilibrium is given in Figure 4. To ensure that the
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dipole moment data used in the calculation is smooth, the RCCSD(T) dipole moment points in the 1.4-5.8 Å range were used in a linear fit, also plotted in Figure 4. The obtained linear function is M(r)= (3.27907-2.28639r), in a.u.. The corresponding adjusted R-squared value is 0.9998. Such a fitting procedure is a simple solution to avoid some unexpected numerical noise present in the original ab initio points. This numerical noise can cause problems when the ab
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initio points are used directly in the calculation of intensities for high overtone transitions [46]. This linear function was used in the computation of line list of CaF for v=0-10, N≤121. The corresponding energy is up to about ¼ of the depth of the potential well; it is also just a small
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part of the linear range of the dipole moment function.
Similar to MgF [44], also a highly ionic molecule, the dipole moment curve of Ca+F- is expected to be affected by a curved crossing with the neutral CaF potential curve and to converge rapidly
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to zero at large nuclear separations. A simple estimate can be made for the location of this curve crossing based on the electron affinity of F and the first ionization potential of Ca [30,47,48]; the estimated potential curve crossing occurs near 5.29 Å (≈2.7 re). Therefore, the ionic character of the molecule may last to this distance and then the dipole moment will start to
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converge towards zero. The various calculations in Figure 2 start to have convergence
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problems around 6 Å depending on the method, the basis set and the electronic configuration. Note that the RCCSD(T) dipole in the range r=6-9 Å is not regular. The MRCI and ACPF data
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confirm the reliability of the RCCSD(T) dipole in the range of from 1.4 Å to about 6 Å, which is
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enough for our line list calculations.
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Table 3. Dipole moments for CaF ground state at re from different methods. Method FD-HF MHF SCF SDCI CPF RCCSD(T) ACPF/(Ca:v5z;F:av5z) ACPF/v5z ACPF/cc-pcvqz
Dipole moment 2.6450 D 2.6451 D 2.659 D 2.59 D 3.06 D 3.05 D 2.83 D 2.81 D 2.77 D
Reference Kobus, et al. 2000 [14] Kobus, et al. 2000 [14]
Langhoff et al. 1986 [8] Langhoff et al. 1986 [8] Langhoff et al. 1986 [8] Harrison et al. (2002) [16] This work a This work a This work a
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ACPF/cc-pcv5z 2.81 D This work a MRCI/cc-pcv5z 2.74 D This work a RCCSD(T) 3.04 D This work, at re=1.9526 Å Exp. 3.05(7) D Childs, et al. 1986 [7] b a Corresponds to the dipole at r=2.0 Å. Active space (6,3,3,1), doubly occupied and core orbitals (6,2,2); in the reference CASSCF wavefunction calculation, active space (6,3,3,1); doubly occupied orbitals (6,2,2); frozen orbitals (4,1,1). Corresponding to μe in v-dependent dipole moment expression μ(v)= μe+μ1(v+½), where μe=3.05(7), μ1=0.052(6), in debye. [7]
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b
Table 4. Calculated ΔGv (in cm-1) from RCCSD(T) and EMO potentials for CaF ground state. 0 1 2 3 4 5 6 7 8 re/Å
ΔGv/Exp.[11] 293.5944 582.8478 577.0991 571.4017 565.7547 560.1581 554.6119 549.1164 543.6705 1.9516
RCCSD(T)
292.4585 580.5510 575.1047 569.7159 564.3234 559.0171 553.7924 548.5669 543.4071 1.9526
EMO 293.6586 582.852 577.1025 571.4028 565.7531 560.1533 554.6036 549.1037 543.6536
Δ/cm-1 0.0642 0.0042 0.0034 0.0011 -0.0016 -0.0048 -0.0083 -0.0127 -0.0169
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4 Line lists
Δ/cm-1 -1.1359 -2.2968 -1.9944 -1.6858 -1.4313 -1.141 -0.8195 -0.5495 -0.2634
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Ro-vibrational line lists were obtained by direct solution of the Schrödinger equation for nuclear motion using the LEVEL program. The intensity of a line as measured by the Einstein A value is given by the expression
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[24,49]:
(
)
|〈
( )
〉|
(4)
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in which A is in s-1 units, M(r) is the dipole moment function in debye, ν the transition frequency in cm-1, S(J',J'') is a Hönl-London rotational factor, and Ψv'J', and Ψv''J'', are the vibrational wavefunctions, which
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depend on the effective molecular potential function Veff(r)=V(r)+Vcent(J), where Vcent(J) is the J-dependent centrifugal potential [49]. From Eq. (4), the two factors that influence the vibration-rotation line intensities in an electronic state are the wavefunctions ΨvJ, determined by the effective potential, and the dipole moment
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function M(r). In this work, we use an accurate fitted EMO potential function for V(r) and a linear fit of the RCCSD(T) dipole moment function (M(r)= 2.541746(3.27907 -2.28639 r); in debye). For the CaF 2Σ+ ground state, the spin-rotation splitting is small and is not observed in the vibration-rotation emission spectrum [11]. CaF was therefore treated as a 1Σ+ state [11] for the purposes of generating vibrationrotation line lists. Note that this actually corresponds to Hund’s case (b) but without fine structure. The main difference between this 2Σ+ state and a 1Σ+ state lies in the statistical weight factor (2S+1) involved in the
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partition function, but does not influence the final population of vibration-rotation levels since the factor is the same in each level. Using the EMO potential for 40CaF and the linear fit of the RCCSD(T) dipole moment, rovibrational line lists were obtained for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF with LeRoy's LEVEL program [24]. As expected, LEVEL produces transitions nearly identical to those from dPotFit for 40CaF. LEVEL also produces transition dipole matrix elements, the corresponding Einstein A values for each line and a set of band constants (Bv, Dv, Hv, …, Ov) for each vibrational level. These band constants are useful for
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simulating spectra with a program such as PGOPHER [50]. Line lists for all possible allowed transitions with v=0–10, N≤121, with ΔN=±1, are calculated for the 40CaF and its isotopologues and are available as
supplementary files (Tables S1-S7). Comparison between calculated and observed transitions of 40CaF is given in Table S1. Results from two sets of vibrational terms Gv are given in Table S1, one was taken from the observed data (v=0-8, with G0 set to zero) given by Charron et al. [11], another is from LEVEL (v=0-10) with the EMO potential given in Table 2. All the other band constants are from LEVEL. The RMS error of the
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calculated transitions with all band constants from LEVEL was 0.0061 cm-1; that from the experimental Gv and LEVEL constants (Bv, Dv, …, Ov) was 0.0017 cm-1. Line lists directly from LEVEL are given in Table S2-S7 for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF, respectively. The band constants are collected and given in Table S8 for all isotopologues. Born-Oppenheimer breakdown effects were ignored in the line list calculations since Ca and F are relatively heavy nuclei. For most of the observed rovibrational transitions of 40CaF, the 1
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differences between the calculated and the observed transitions from Charron et al. [11] are less than 0.005 cm.
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The calculated emission line intensities, I, for 40CaF (at 1873 K) are presented in Figure 5 as an overview and in Figure 6 for a short segment of the observed spectrum [11]. These intensities (in units of s-1/molecule) are
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estimated as relative photon emission rates [49]: (
),
(5)
-1
in which Q is the partition function, A is the Einstein A coefficient (s ) and gupper is the upper state degeneracy.
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As expected, the strongest spectra appear in the 400–650 cm-1 range, corresponding to the Δv=1 transitions. For the overtone bands, the intensities decrease by a factor of about 10 for each increase in Δv by one unit.
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Intensity anomalies [51] like those calculated for the overtone spectra of MgF [44] did not appear for CaF. As a test of our line lists, we simulated the emission spectrum of 40CaF using PGOPHER for comparison with the observed spectrum of Charron et al. [11] recorded at a sample temperature of 1873 K. The positions and relative intensities agree well as shown in Figure 6.
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5. Conclusion Line lists are calculated for the ground state for CaF isotopologues with an accurate empirical EMO potential function which reproduces the observed transitions within experimental uncertainty. Ab initio dipole moment and molecular potential functions are investigated with MRCI, ACPF, and RCCSD(T) methods. An accurate linear analytical dipole moment function, which matches the observed value at equilibrium, was obtained by fitting the RCCSD(T) data. The line intensities predicted with this dipole moment function are expected to were predicted for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF.
Acknowledgements
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give reliable estimates for intensities of overtone transitions. All transitions for v=0–10, N≤121, with ΔN=±1,
We thank Dr Xiaoguang Ma (Ludong University) for performing the MRCI/vqz calculation on their
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supercomputer. This work was supported by the China Scholarship Council, Shandong Province Natural Science Foundation (ZR2015AM009). Some support to PFB was provided by the NASA Exoplanet
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program.
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[18] Field RW, Gittins CM, Harris NA, and Jungen Ch. Quantum defect theory of dipole and vibronic mixing in Rydberg states of CaF. J Chem Phys 2005;122:184314. [19] Yang CL, Zhang XY, Gao F, Ren TQ. MRCI study on the potential energy curves and
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spectroscopic parameters of the ground and low-lying excited states in CaF. J Mol Struct: THEOCHEM 2007;807:147–152. [20] Kay JJ, Altunata SN, Coy SL, Field RW. Resonance between electronic and rotational motions in Rydberg states of CaF. Mol Phys 2007;105:1661-73.
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energy functions. J Quant Spectrosc Rad Transfer 2017;186:179–196. [26] Le Roy RJ. Determining Equilibrium Structures and Potential Energy Functions for Diatomic
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Knizia G, Manby FR, Schütz M, Celani P, Korona T, Lindh R, Mitrushenkov A, Rauhut G, Shamasundar KR, Adler TB, Amos RD, Bernhardsson A, Berning A, Cooper DL, Deegan MJO, Dobbyn AJ, Eckert F, Goll E, Hampel C, Hesselmann A, Hetzer G, Hrenar T, Jansen G, Köppl C, Liu Y, Lloyd AW, Mata RA, May AJ, McNicholas SJ, Meyer W, Mura ME,
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Nicklass A, O'Neill DP, Palmieri P, Peng D, Pflüger K, Pitzer R, Reiher M, Shiozaki T, Stoll H, Stone AJ, Tarroni R, Thorsteinsson T, Wang M. See http://www.molpro.net; 2012.
[29] Li G, Gordon IE, LeRoy RJ, Hajigeorgiou PG, Coxon JA, Bernath PF, Rothman LS.
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Reference spectroscopic data for hydrogen halides. Part I: construction and validation of the ro-vibrational dipole moment functions. J Quant Spectrosc Rad Transfer 2013;121:78–90. [30] Haynes WM, editor. CRC Handbook of Chemistry and Physics. 95th edition. Boca Raton, FL.: CRC press; 2014.
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[31] Werner H-J, Knowles PJ. An efficient internally contracted multiconfiguration-reference
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[35] Werner H-J, Knowles PJ. A comparison of variational and non-variational internally contracted multiconfiguration-reference configuration interaction calculations. Theor Chim Acta 1990; 78:175-187.
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[36] Knowles PJ, Hampel C, Werner H-J. Coupled cluster theory for high spin, open shell reference wave functions. J Chem Phys 1993; 99:5219-5227; Erratum: J Chem Phys 2000; 112:3106-3107. [37] Dunning TH. Gaussian basis sets for use in correlated molecular calculations. I. The atoms
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[38] Kendall RA, Dunning Jr TH, Harrison RJ. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J Chem Phys 1992;96:6796–6806.
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[39] Woon DE, Dunning Jr TH. Gaussian basis sets for use in correlated molecular calculations. V. Core-valence basis sets for boron through neon. J Chem Phys 1995;103:4572–4585. [40] Wilson AK, van Mourik T, Dunning Jr TH. Gaussian basis sets for use in correlated molecular calculations. VI. Sextuple zeta correlation consistent basis sets for boron through
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neon. J Mol Struct: THEOCHEM 1996;388:339–349.
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[41] Koput J, Peterson KA. Ab Initio Potential Energy Surface and Vibrational-Rotational Energy Levels of X2Σ+ CaOH. J Phys Chem A 2002; 106:9595-9599.
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[42] Knowles PJ, Werner H-J. An efficient second-order MC SCF method for long configuration expansions. Chem Phys Lett 1985;115(3):259–267.
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convergence. J Chem Phys 1985;82:5053–5063. [44] Hou S, Bernath PF. Line list for the MgF ground state. J Quant Spectrosc Rad Transfer 2017; 203: 511–516.
[45] Li G, Gordon IE, Rothman LS, Tan Y, Hu S-M, Kassi S, Campargue A, Medvedev ES. Rovibrational line lists for nine isotopologues of the CO molecule in the X1Σ+ ground electronic state. Astrophys J Supp 2015; 216:15 (18pp)
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[46] Medvedev ES, Meshkov VV, Stolyarov AV, Ushakov VG, Gordon IE. Impact of the dipolemoment representation on the intensity of high overtones. J Mol Spectrosc 2016;330:36–42. [47] Jakubek ZJ, Harris NA, Field RW, Gardner JA, Murad E. Ionization potentials of CaF and BaF. J Chem Phys 1994;100:622-627.
2nd ed. (Van Nostrand Princeton NJ 1950).
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[48] Herzberg G. Molecular Spectra and Molecular Structure. I Spectra of Diatomic Molecules.
[49] Bernath PF. Spectra of Atoms and Molecules. 3rd ed. New York: Oxford University Press;
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[50] Western CM. PGOPHER: a program for simulating rotational, vibrational and electronic spectra. J Quant Spectrosc Rad Transfer 2017;186:221–242.
[51] Medvedev ES, Ushakov VG, Stolyarov AV, Gordon IE. Intensity anomalies in the rotational
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and ro-vibrational spectra of diatomic molecules. J Chem Phys 2017;147:164309.
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Figure 1. Ab initio molecular potential functions for the CaF ground state. In the RCCSD(T) calculation
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(dash dotted (“·”)), all the electrons were correlated. In ACPF or MRCI calculations all core orbitals (6,2,2) were doubly occupied and frozen, and the active space was {6,3,3,1} for the remaining 9 valence electrons. CASSCF wavefunctions were used as a reference for ACPF and MRCI calculations. If the
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ACPF/MRCI space configurations are the same as the reference configurations, the double occupied and frozen/core spaces are given in parentheses, respectively, i.e., (622,622) and (622,411), following the
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method/basis labels (indicated with plus (“+”), triangle (“△”), and stars (“*”)). Otherwise, the doubly occupied space was (6,2,2); the frozen space for in CASSCF was (4,1,1), while in the following ACPF
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and MRCI calculations the core orbitals were still set to (6,2,2); the corresponding results were indicated with circles (“○”), crosses (“x”), and solid (“―”), dashed (“”), and dotted (“…”) lines; basis sets were “vxz”, “pcvxz”, and “avxz” indicate “cc-pvxz”, “cc-pcvxz”, and “aug-cc-pvxz” (x=q,5), respectively.
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Figure 2. Ab initio dipole moment functions for the CaF ground state. As in Figure 1, but for dipole
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strength of 0.0005 a.u.
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moment functions. In the RCCSD(T) calculation, the finite field approach was adopted with a field
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electrons were correlated.
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Figure 3. RCCSD(T) and EMO potentials for the CaF ground state. In the RCCSD(T) calculation, all the
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Figure 4. Dipole moment in the linear range from some ACPF and RCCSD(T) calculations. ACPF configurations have a (6,3,3,1) active space, with doubly occupied and core orbitals set to (6,2,2);
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in the reference CASSCF calculation, the active space and doubly occupied orbitals are the same as those for ACPF, but the frozen orbitals are (4,1,1). In the RCCSD(T) calculations all electrons were correlated.
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The linear function M(r) (solid line “―”) was obtained by fitting the RCCSD(T) data (dashed line “- -”) in the range of r=1.4-5.8 Å. Basis sets have same meaning as those in Figure 1. Observed data point (“+”)
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was taken from Childs et al. 1986 [7].
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Figure 5. Overall intensities for transitions at 1873 K in the range of v=0-10, N≤121 with ∆v=0-10, ∆
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N=±1.
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Figure 6. Comparison with a portion of observed (upper; in black) R-branch spectrum and the simulated (lower; in blue) results with PGOPHER [50] using band constants and TDMs from LEVEL using the EMO potential and RCCSD(T) dipole of the present work. The observed spectrum was taken from
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Charron et al. [11]. The temperature for the simulated spectrum was 1873 K. The vertical scale is for the
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simulated spectrum.
Line list for the ground state of CaF Shilin Hou , Peter F. Bernath PII: DOI: Reference:
S0022-4073(17)30980-9 10.1016/j.jqsrt.2018.02.011 JQSRT 5992
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
29 December 2017 7 February 2018 7 February 2018
Please cite this article as: Shilin Hou , Peter F. Bernath , Line list for the ground state of CaF, Journal of Quantitative Spectroscopy & Radiative Transfer (2018), doi: 10.1016/j.jqsrt.2018.02.011
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Highlights Analytical Extended Morse Oscillator (EMO) potential function derived for CaF
Ground state dipole moment function calculated ab initio
Line lists calculated for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF vibration-rotation transitions
Useful for spectra of high temperature environments such as hot super-Earth exoplanets
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Line list for the ground state of CaF Shilin Hou a and Peter F. Bernath b,* a b
Department of Physics, Ocean University of China, Qingdao, Shandong 266100, China
Department of Chemistry and Biochemistry, Old Dominion University, Norfolk, Virginia 23529, USA
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*Corresponding author. E-mail address: [email protected] (P.F. Bernath).
Abstract
The molecular potential energy function and electronic dipole moment function for the ground state of
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CaF were studied with MRCI, ACPF, and RCCSD(T) ab initio calculations. The RCCSD(T) potential function reproduces the experimental vibrational intervals to within ~2 cm-1. The RCCSD(T) dipole moment at the equilibrium internuclear separation agrees well with the experimental value. Over a wide range of internuclear separations, far beyond the range associated with the observed spectra, the ab initio dipole moment functions are similar and highly linear. An extended Morse oscillator (EMO) potential function was also obtained by fitting the observed lines of the laboratory vibration-rotation and pure
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rotation spectra of the 40CaF X2Σ+ ground state. The fitted potential reproduces the observed transitions (v≤8, N≤121, Δv=0, 1) within their experimental uncertainties. With this EMO potential and the
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RCCSD(T) dipole moment function, line lists for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF, and 48CaF were computed for v≤10, N≤121, Δv=0–10. The calculated emission spectra are in good agreement with an
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observed laboratory spectrum of CaF at a sample temperature of 1873 K.
Keywords: Molecular potential; dipole moment function; Einstein A coefficients; vibration-rotation line
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lists
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1 Introduction Calcium fluoride is a molecule of interest in many fields such as astrophysics, spectroscopy, computational chemistry, and in cold molecule trapping [e.g., 1-21]. Optical, infrared, and microwave spectra have been obtained and analyzed for transitions associated with the ground state and many excited electronic states [2-5, 9-13]. For example, electronic transitions involving the A2Π，B2Σ+, C2Π,
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E2Σ+ and E'2Π states [2-5], and many Rydberg states [18, 20] are known. Recently the CaF B-X system was used to improve the analytical capabilities of LIBS (laser-induced breakdown spectroscopy) fluorine detection in Ca-containing samples as well as in calcium-free samples [22].
For the ground state, vibration-rotation lines for v=0–8 and N up to 121 were analyzed by Charron et al. [11] from an infrared emission spectrum; millimeter-wave transitions were observed by Anderson et al.
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[9] for v=0–3, N from 3 to 18. Stark splittings were studied by Childs et al. [6,7] for v=0-1, from which the dipole moments for v=0 (3.07(7) debye; D) and v=1 (3.12(7) D) were obtained; the dipole moment at the equilibrium internuclear separation was derived as 3.05(7) D.
In circumstellar environments, such around the cool carbon star IRC+10216, metal salts containing
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sodium and aluminum, such as AlCl, AlF, and NaCl, have already been detected [23]. Because the cosmic abundance of calcium is similar to that of aluminum and sodium, other metal fluorides such as CaF may
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be detectable also [10].
As a free radical with a single electron outside closed-shell ionic cores, CaF is also of special theoretical
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significance [8,12,14-20] especially for studies of the properties of its Rydberg states [12,18,20]. For the ground state, highly ionic bonding (Ca+F-) near equilibrium is well-known and is supported by the
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relatively small value of the hyperfine constant b, which includes the Fermi contact electron spin density term [1]. This implies that the electron density at the F nucleus (which has the nuclear spin) is not very
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large as expected if the lone unpaired electron is primarily located on the metal atom.
The purpose of the present work is to prepare a CaF vibration-rotation line list with positions and line strengths. In order to obtain line strengths, a dipole moment function is computed with high quality ab initio methods. The CaF line list can be used for quantitative analysis and prediction of infrared spectra of hot CaF as found for example in hot super-Earth exoplanets. The ro-vibrational line positions (and line intensities) are obtained by direct solution of the Schrödinger
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equation for nuclear motion using Le Roy's computer program LEVEL [24] with an accurate empirical potential energy function. The potential energy function was obtained by fitting the pure rotation [9] and vibration-rotation [11] data using LeRoy's dPotFit code [25]. The potential function used was an Expanded Morse Oscillator (EMO) [26] similar to the work of Barton et al. [27] on NaCl and KCl.
The ground state potential energy function was studied with both ab initio and empirical approaches. The
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ab initio data were obtained to study the behavior over the full range of the potential; the empirical
potential was used to improve the accuracy of the predicted vibration-rotation transitions. All ab initio computations were performed with the Molpro [28] package.
The dipole moment at the equilibrium internuclear separation, re, which can be derived from the
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experimental data for the v=0 and 1 levels, was used as an important criterion to choose the best dipole moment function from the ab initio computations. The calculated line intensities are very sensitive to the form of the dipole moment function and its derivatives [29]; therefore, dipole moment functions from several different ab initio approaches were compared, and the best one was then used in the LEVEL program to calculate the line intensities for 40CaF and its isotopologues. This calculation also gives a high
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quality molecular potential function which best reproduces the observed spectroscopic data.
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2 Fitted molecular potential from observed spectra The line positions of CaF can be reproduced using Le Roy’s extended Morse oscillator (EMO) potential
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model [26], which can be expressed as
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where
( )(
( )
( )
∑
(
)
)
(
(1)
) .
(2) (3)
De is the equilibrium potential well depth and re is the equilibrium internuclear distance. N and p are preset parameters, and the i parameters are determined using Le Roy’s potential fitting program dPotFit [25]. p was set to 3, N was set to 5, De was fixed and re was set free in the fit. Information for the observed transitions adopted for fitting is listed in Table 1.
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De was set to 44203.50 cm−1, which was obtained using De=D0+E0, with D0 taken from Haynes [30], and the zero-point energy E0 calculated from the spectroscopic constants of Charron et al. [11]. In total 811 observed transitions were fitted with a dimensionless standard error (DSE) of 0.89. The 40CaF pure rotation data were corrected for the effects of fine and hyperfine structure as in Charron et al. [11]. For the vibration-rotation lines, the uncertainties assigned in the original experimental work [11], in the range of 0.001 cm−1 − 0.005 cm−1, were used and for the millimeter-wave data [9], an uncertainty of 9.3×10−7
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cm−1 was adopted. These uncertainties correspond to one standard deviation. The small spin splitting and F hyperfine structure was not resolved in the vibration-rotation spectrum [ 11] recorded at 0.01 cm−1 resolution and CaF was treated as if the ground state has 1Σ+ symmetry. In the present work, we aim to provide a vibration-rotation line list so we follow Charron et al. [11] and ignore fine and hyperfine
structure. The EMO potential parameters obtained from dPotFit are presented in Table 2 along with 95%
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confidence limits.
Table 1. Laboratory data used to determine the CaF potential energy curve. Transitions CaFJ ∆N N
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Charron, et al. 1995 [11] T=1873 K
∆N
Uncertainty (cm-1)
1.7x10-5 cm-1
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Table 2. EMO potential parameters for CaF. CaF EMO(N=5, p=3) De/cm-1 44203.50 re/Å 1.951636551397 1.223498191939D+00 -2.618924324279D-01 -6.058869340934D-02 5.708544137802D-02 -5.248856409990D-02 7.362231681793D-02
Uncertainty Fixed 2.5D-07 1.0D-06 2.2D-05 9.3D-05 8.2D-04 2.7D-03 1.1D-02
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0.001-0.005 cm-1
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Anderson, et al. 1994 [9]
Frequency range (cm-1)
3 Ab initio dipole moment functions and molecular potentials To obtain a reliable dipole moment function for calculating transition dipole moments, several high-level ab initio calculations were performed with various large basis sets. Methods used include the internallycontracted multiconfiguration reference configuration interaction (MRCI) [31-33], the averaged coupledpair functional approach (ACPF) [34-35], and the partially spin restricted open-shell coupled cluster theory with single and double excitations plus a correction for triple excitations (RCCSD(T)) [36].
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Correlation-consistent basis sets [37-41] were used in the calculations. Basis functions mainly include ccpvxz, cc-pcvxz for Ca and/or F, and aug-cc-pvxz for F with x=Q and 5 [37-41]. All calculations were performed with Molpro, using the basis sets embedded in the package.
For the ground state of CaF, the 29 electrons are distributed in 15 molecular orbitals, with orbital energy
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order at the equilibrium geometry 1σ22σ23σ24σ21π45σ26σ22π47σ23π48σ29σ1. The electronic configuration can be expressed as (9a1, 3b1, 3b2, 0a2) (more compactly (9,3,3,0) or (9,3,3)) using the irreducible
representations of the C2v point group used by Molpro. The wavefunctions from state-averaged complete active space self-consistent field (CASSCF) [42,43] calculations were used as reference wavefunctions for the MRCI and ACPF calculations. In most of the MRCI and ACPF calculations, the active space, the doubly occupied space, and frozen/core space are the same as that of the corresponding CASSCF
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calculations. When the occupied orbitals were set to be (9,3,3) and (10,4,4) (in which the s and p atomic orbitals were included in the reference), the doubly occupied orbitals were set to be (6,2,2) or (4,1,1) and the frozen orbitals were set to be the same as the doubly occupied space or smaller, the obtained potential curves and the dipole moment functions were smooth, but the computed dipole moment functions were not converged at large distances. The ACPF and MRCI dipole moments obtained were about ~10% lower
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than the observed value at the equilibrium geometry.
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MgF (dipole moment ~3.004 D) [44] and CaF (~3.05 D) have similar dipole moments at equilibrium; therefore, some similarity in the dipole moment function is expected. To obtain a converged dipole moment function at large distances, the active space was enlarged, and d atomic orbitals (of Ca) were
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included in the active space, i.e., the occupied orbitals were set to be (12,5,5,1). The doubly occupied space was (6,2,2); thus, the active space was set as {6,3,3,1} for the remaining 9 valence electrons in the
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corresponding CASSCF/ACPF (or MRCI) calculations. Some typical results are shown in Figures 1 and 2
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for comparison.
When the doubly occupied, frozen core space was (6,2,2), converged dipole moment functions were obtained; but the calculated potential curves had a second minimum at about 2.5-3.0 Å (shown with “+” and “△” symbols in Figure 1), which is not physically correct. To solve this problem, the frozen/core spaces in MRCI, ACPF, and CASSCF computations were set to be
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(4,1,1). The cc-pvqz basis sets were tried for both Ca and F. Note that the cost for such a calculation is large at present. For example, in the MRCI/cc-pvqz calculation, there are in total 107,694 configuration state functions (CSFs) involved in the reference space, and 64,990,638 internal contracted CSFs involved in the MRCI configuration space. After more than one month of
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computation on a supercomputer, only 13 points in the range of 1.4-3.9 Å were obtained (shown with “*” symbols in Figure 1). As expected, the second minimum disappeared, and both the potential curve and the dipole moment function are smooth as well. To get a more economical solution, we tried a mixed scheme, i.e., in the CASSCF calculation, the frozen space was set to
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be (4,1,1), while in the following ACPF (or MRCI) calculations, the core spaces were set to be (6,2,2). The calculated ACPF and MRCI potential are smooth, and the dipole moment functions are similar to that obtained with the frozen/core spaces both set to (4,1,1). Considerable
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computation time was saved in this way. However, the MRCI and ACPF dipole moments were
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not improved near equilibrium compared to the observed data [6].
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The comparison between dipole moments from difference methods is given in Table 3. In this table, literature results with CPF (coupled pair functional) [8] and RCCSD(T) [16] approaches gave the predictions of 3.06 D [8] and 3.05 D [16], respectively, which are nearly identical to our RCCSD(T)
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prediction (3.04 D). These predictions are close to the experimental equilibrium dipole moment of 3.05(7) D given by Childs et al. [7]. Most of the other predictions in Table 3 differ by relatively large amounts
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(>10%) from this value.
In our RCCSD(T) calculations, different basis set were also tried; finally, we found that the RCCSD(T) approach with basis sets for Ca=cc-pv5z and F=aug-cc-pv5z gave the best dipole as well as molecular potential in the range of observed data. The dipole moment points were
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obtained with the finite field method embedded in Molpro, with the electric field set to the small value of 0.0005 a.u. (see for example, Ref. [45]). The obtained dipole moment at the optimized re (1.9526 Å) is 3.04 D (1.1947 a.u.), which agrees well with the observed value 3.05(7) D [7]. This RCCSD(T) re value also agrees well with the experimental value of re=1.9516 Å [11]. When the
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potential well is shifted to the EMO potential, the RCCSD(T) potential agrees very well with the EMO potential. A comparison between the EMO potential and the RCCSD(T) data is shown in Figure 3. The calculated ∆Gv values from this RCCSD(T) potential agree well with the experimental data (differences ~2 cm-1), as presented in Table 4. The largest relative
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discrepancy is less than 0.4%. The RCCSD(T) dipole moment function is plotted in Figure 2.
The results in Figure 2 show that some of the dipole moment functions do not converge. The
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dipole moment functions are very linear up to about 6 Å (except for ACPF/cc-pcvqz) for MRCI/ACPF with relatively large basis sets, which is far large than the range (less than 3 Å)
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sampled by the observed transitions of Charron et al. [11]. Since the RCCSD(T) approach gives
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the best value compared to the observed value at equilibrium, and converged to the high-level
ab inito results (e.g., ACPF/(Ca:v5z;F:av5z) and MRCI/cc-pcv5z in Figure 2 ) at large distances,
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the RCCSD(T) dipole moment function was used for the line intensity calculation. A comparison between dipole moment functions around equilibrium is given in Figure 4. To ensure that the
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dipole moment data used in the calculation is smooth, the RCCSD(T) dipole moment points in the 1.4-5.8 Å range were used in a linear fit, also plotted in Figure 4. The obtained linear function is M(r)= (3.27907-2.28639r), in a.u.. The corresponding adjusted R-squared value is 0.9998. Such a fitting procedure is a simple solution to avoid some unexpected numerical noise present in the original ab initio points. This numerical noise can cause problems when the ab
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initio points are used directly in the calculation of intensities for high overtone transitions [46]. This linear function was used in the computation of line list of CaF for v=0-10, N≤121. The corresponding energy is up to about ¼ of the depth of the potential well; it is also just a small
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part of the linear range of the dipole moment function.
Similar to MgF [44], also a highly ionic molecule, the dipole moment curve of Ca+F- is expected to be affected by a curved crossing with the neutral CaF potential curve and to converge rapidly
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to zero at large nuclear separations. A simple estimate can be made for the location of this curve crossing based on the electron affinity of F and the first ionization potential of Ca [30,47,48]; the estimated potential curve crossing occurs near 5.29 Å (≈2.7 re). Therefore, the ionic character of the molecule may last to this distance and then the dipole moment will start to
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converge towards zero. The various calculations in Figure 2 start to have convergence
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problems around 6 Å depending on the method, the basis set and the electronic configuration. Note that the RCCSD(T) dipole in the range r=6-9 Å is not regular. The MRCI and ACPF data
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confirm the reliability of the RCCSD(T) dipole in the range of from 1.4 Å to about 6 Å, which is
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enough for our line list calculations.
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Table 3. Dipole moments for CaF ground state at re from different methods. Method FD-HF MHF SCF SDCI CPF RCCSD(T) ACPF/(Ca:v5z;F:av5z) ACPF/v5z ACPF/cc-pcvqz
Dipole moment 2.6450 D 2.6451 D 2.659 D 2.59 D 3.06 D 3.05 D 2.83 D 2.81 D 2.77 D
Reference Kobus, et al. 2000 [14] Kobus, et al. 2000 [14]
Langhoff et al. 1986 [8] Langhoff et al. 1986 [8] Langhoff et al. 1986 [8] Harrison et al. (2002) [16] This work a This work a This work a
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ACPF/cc-pcv5z 2.81 D This work a MRCI/cc-pcv5z 2.74 D This work a RCCSD(T) 3.04 D This work, at re=1.9526 Å Exp. 3.05(7) D Childs, et al. 1986 [7] b a Corresponds to the dipole at r=2.0 Å. Active space (6,3,3,1), doubly occupied and core orbitals (6,2,2); in the reference CASSCF wavefunction calculation, active space (6,3,3,1); doubly occupied orbitals (6,2,2); frozen orbitals (4,1,1). Corresponding to μe in v-dependent dipole moment expression μ(v)= μe+μ1(v+½), where μe=3.05(7), μ1=0.052(6), in debye. [7]
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b
Table 4. Calculated ΔGv (in cm-1) from RCCSD(T) and EMO potentials for CaF ground state. 0 1 2 3 4 5 6 7 8 re/Å
ΔGv/Exp.[11] 293.5944 582.8478 577.0991 571.4017 565.7547 560.1581 554.6119 549.1164 543.6705 1.9516
RCCSD(T)
292.4585 580.5510 575.1047 569.7159 564.3234 559.0171 553.7924 548.5669 543.4071 1.9526
EMO 293.6586 582.852 577.1025 571.4028 565.7531 560.1533 554.6036 549.1037 543.6536
Δ/cm-1 0.0642 0.0042 0.0034 0.0011 -0.0016 -0.0048 -0.0083 -0.0127 -0.0169
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4 Line lists
Δ/cm-1 -1.1359 -2.2968 -1.9944 -1.6858 -1.4313 -1.141 -0.8195 -0.5495 -0.2634
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v
Ro-vibrational line lists were obtained by direct solution of the Schrödinger equation for nuclear motion using the LEVEL program. The intensity of a line as measured by the Einstein A value is given by the expression
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[24,49]:
(
)
|〈
( )
〉|
(4)
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in which A is in s-1 units, M(r) is the dipole moment function in debye, ν the transition frequency in cm-1, S(J',J'') is a Hönl-London rotational factor, and Ψv'J', and Ψv''J'', are the vibrational wavefunctions, which
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depend on the effective molecular potential function Veff(r)=V(r)+Vcent(J), where Vcent(J) is the J-dependent centrifugal potential [49]. From Eq. (4), the two factors that influence the vibration-rotation line intensities in an electronic state are the wavefunctions ΨvJ, determined by the effective potential, and the dipole moment
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function M(r). In this work, we use an accurate fitted EMO potential function for V(r) and a linear fit of the RCCSD(T) dipole moment function (M(r)= 2.541746(3.27907 -2.28639 r); in debye). For the CaF 2Σ+ ground state, the spin-rotation splitting is small and is not observed in the vibration-rotation emission spectrum [11]. CaF was therefore treated as a 1Σ+ state [11] for the purposes of generating vibrationrotation line lists. Note that this actually corresponds to Hund’s case (b) but without fine structure. The main difference between this 2Σ+ state and a 1Σ+ state lies in the statistical weight factor (2S+1) involved in the
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partition function, but does not influence the final population of vibration-rotation levels since the factor is the same in each level. Using the EMO potential for 40CaF and the linear fit of the RCCSD(T) dipole moment, rovibrational line lists were obtained for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF with LeRoy's LEVEL program [24]. As expected, LEVEL produces transitions nearly identical to those from dPotFit for 40CaF. LEVEL also produces transition dipole matrix elements, the corresponding Einstein A values for each line and a set of band constants (Bv, Dv, Hv, …, Ov) for each vibrational level. These band constants are useful for
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simulating spectra with a program such as PGOPHER [50]. Line lists for all possible allowed transitions with v=0–10, N≤121, with ΔN=±1, are calculated for the 40CaF and its isotopologues and are available as
supplementary files (Tables S1-S7). Comparison between calculated and observed transitions of 40CaF is given in Table S1. Results from two sets of vibrational terms Gv are given in Table S1, one was taken from the observed data (v=0-8, with G0 set to zero) given by Charron et al. [11], another is from LEVEL (v=0-10) with the EMO potential given in Table 2. All the other band constants are from LEVEL. The RMS error of the
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calculated transitions with all band constants from LEVEL was 0.0061 cm-1; that from the experimental Gv and LEVEL constants (Bv, Dv, …, Ov) was 0.0017 cm-1. Line lists directly from LEVEL are given in Table S2-S7 for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF, respectively. The band constants are collected and given in Table S8 for all isotopologues. Born-Oppenheimer breakdown effects were ignored in the line list calculations since Ca and F are relatively heavy nuclei. For most of the observed rovibrational transitions of 40CaF, the 1
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differences between the calculated and the observed transitions from Charron et al. [11] are less than 0.005 cm.
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The calculated emission line intensities, I, for 40CaF (at 1873 K) are presented in Figure 5 as an overview and in Figure 6 for a short segment of the observed spectrum [11]. These intensities (in units of s-1/molecule) are
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estimated as relative photon emission rates [49]: (
),
(5)
-1
in which Q is the partition function, A is the Einstein A coefficient (s ) and gupper is the upper state degeneracy.
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As expected, the strongest spectra appear in the 400–650 cm-1 range, corresponding to the Δv=1 transitions. For the overtone bands, the intensities decrease by a factor of about 10 for each increase in Δv by one unit.
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Intensity anomalies [51] like those calculated for the overtone spectra of MgF [44] did not appear for CaF. As a test of our line lists, we simulated the emission spectrum of 40CaF using PGOPHER for comparison with the observed spectrum of Charron et al. [11] recorded at a sample temperature of 1873 K. The positions and relative intensities agree well as shown in Figure 6.
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5. Conclusion Line lists are calculated for the ground state for CaF isotopologues with an accurate empirical EMO potential function which reproduces the observed transitions within experimental uncertainty. Ab initio dipole moment and molecular potential functions are investigated with MRCI, ACPF, and RCCSD(T) methods. An accurate linear analytical dipole moment function, which matches the observed value at equilibrium, was obtained by fitting the RCCSD(T) data. The line intensities predicted with this dipole moment function are expected to were predicted for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF and 48CaF.
Acknowledgements
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give reliable estimates for intensities of overtone transitions. All transitions for v=0–10, N≤121, with ΔN=±1,
We thank Dr Xiaoguang Ma (Ludong University) for performing the MRCI/vqz calculation on their
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supercomputer. This work was supported by the China Scholarship Council, Shandong Province Natural Science Foundation (ZR2015AM009). Some support to PFB was provided by the NASA Exoplanet
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program.
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Figure 1. Ab initio molecular potential functions for the CaF ground state. In the RCCSD(T) calculation
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(dash dotted (“·”)), all the electrons were correlated. In ACPF or MRCI calculations all core orbitals (6,2,2) were doubly occupied and frozen, and the active space was {6,3,3,1} for the remaining 9 valence electrons. CASSCF wavefunctions were used as a reference for ACPF and MRCI calculations. If the
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ACPF/MRCI space configurations are the same as the reference configurations, the double occupied and frozen/core spaces are given in parentheses, respectively, i.e., (622,622) and (622,411), following the
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method/basis labels (indicated with plus (“+”), triangle (“△”), and stars (“*”)). Otherwise, the doubly occupied space was (6,2,2); the frozen space for in CASSCF was (4,1,1), while in the following ACPF
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and MRCI calculations the core orbitals were still set to (6,2,2); the corresponding results were indicated with circles (“○”), crosses (“x”), and solid (“―”), dashed (“”), and dotted (“…”) lines; basis sets were “vxz”, “pcvxz”, and “avxz” indicate “cc-pvxz”, “cc-pcvxz”, and “aug-cc-pvxz” (x=q,5), respectively.
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Figure 2. Ab initio dipole moment functions for the CaF ground state. As in Figure 1, but for dipole
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strength of 0.0005 a.u.
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moment functions. In the RCCSD(T) calculation, the finite field approach was adopted with a field
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electrons were correlated.
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Figure 3. RCCSD(T) and EMO potentials for the CaF ground state. In the RCCSD(T) calculation, all the
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Figure 4. Dipole moment in the linear range from some ACPF and RCCSD(T) calculations. ACPF configurations have a (6,3,3,1) active space, with doubly occupied and core orbitals set to (6,2,2);
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in the reference CASSCF calculation, the active space and doubly occupied orbitals are the same as those for ACPF, but the frozen orbitals are (4,1,1). In the RCCSD(T) calculations all electrons were correlated.
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The linear function M(r) (solid line “―”) was obtained by fitting the RCCSD(T) data (dashed line “- -”) in the range of r=1.4-5.8 Å. Basis sets have same meaning as those in Figure 1. Observed data point (“+”)
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was taken from Childs et al. 1986 [7].
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Figure 5. Overall intensities for transitions at 1873 K in the range of v=0-10, N≤121 with ∆v=0-10, ∆
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Figure 6. Comparison with a portion of observed (upper; in black) R-branch spectrum and the simulated (lower; in blue) results with PGOPHER [50] using band constants and TDMs from LEVEL using the EMO potential and RCCSD(T) dipole of the present work. The observed spectrum was taken from
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Charron et al. [11]. The temperature for the simulated spectrum was 1873 K. The vertical scale is for the
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simulated spectrum.