One and two electrons pseudo-potential investigation of the (FrCs)⁺ and FrCs systems

Abstract

In this study, the potential energy curves (PECs) and dipole moments for ^1,3Σ⁺, ^1,3Π and ^1,3Δ states of the molecule FrCs and for ²Σ⁺, ²Π and ²Δ of (FrCs)⁺ have been computed using a quantum chemistry procedure. This method is based on pseudo-potentials for the representation of atomic core, effective core polarization potential, and large Gaussian basis sets. Besides, we have been deduced from these curves the vibrational levels and the spacing's for all symmetries. The (FrCs)⁺ and FrCs are modeled as one and two valence electrons, respectively and the Fr⁺ and Cs⁺ core are indicated by a pseudo-potential with middle relativistic effects together with the potential of effective core polarization. Since, no experimental results are available for these systems, we have compared our result with the theoretical result found by Aymar et al. and found a good agreement.

Keywords:

• Spectroscopy study
• ab initio
• electronic properties
• vibrational level

1. Introduction

The recent development (Sprouse, Fliller, Grossman, Orozco, & Pearson, 2002) of trapping of radioactive and laser cooling atoms opens ways for new investigations as the research for the β decay, electric dipole moment (EDM), Bose–Einstein condensation, cold atom–atom collisions and more precise atomic clocks.

Laser cooling was first discovered on trapped ions (Neuhauser, Hohenstatt, Toschek, & Dehmelt, 1978; Wineland, Drullinger, & Walls, 1978), which has related to the development of major fields of research. Example of such research is the improvement of spectroscopy and frequency standards (Wineland, 1984), tests of spatial anisotropy (Prestage, Bollinger, Itano, & Wineland, 1985), millikelvin ionic temperatures (Bergquist, Itano, & Wineland, 1987; Nagourney, Janik, & Dehmelt, 1983); confinement of atoms to less than an optical wave length (Bergquist, Hulet, Itano, & Wineland, 1986; Bergquist et al., 1987; Janik, Nagourney, & Dehmelt, 1985; Nagourney, Sandberg, & Dehmelt, 1986; Sauter, Neuhauser, Blatt, & Toschek, 1986) examination of quantum jumps on single ions (Nagourney et al., 1986) and Doppler-free, recoilless spectroscopy (Bergquist et al., 1987; Nagourney et al., 1986). The laser cooling of ions is considerably enabled as their charge allows the ions to be trapped in deep wells. This permits long interaction times and the movement of numerous photon momentum. Moreover, the alkali metals have lately attracted increased attention both experimentally and theoretically because of their importance in the creation of cold molecules (Aymar, Dulieu, & Spiegelman, 2006; Comparat, Drag, Fioretti, Dulieu, & Pillet, 1999; Drag et al., 2000; Dulieu, Raoult, & Tiemann, 2006; Koch, Luc-Koenig, & Masnou-Sweeuws, 2008; Vanhaecke et al., 2004). The component francium has an atomic number Z = 87 and does not have a stable isotope. Francium has a life time of 21.8 min.

The object of this study is to determine the dipolar, vibrational and electronic properties of the FrCs and (FrCs)⁺ systems including: (i) the potential energy curves of various states and the spectroscopic properties (R_e: the equilibrium distance, D_e: well depth, T_e: the transition energies, B_e: rotational constant, ω_e vibrational constant and ω_eχ_e Anharmonic constant). (ii) The permanent dipole moment (PDM) and transition dipole moment (TDM) functions. (iii) The vibrational properties.

Aymar et al. (2006) have been calculated in 2006 the electronic structure of FrCs molecule and its cation. Their investigation has been realized via configuration interaction (CI) approach based on the effective core potential, as the core–valence impact.

This study has been defined as follows: In Section 2, we illustrate the theoretical approach based on adiabatic ab initio investigation. In Section 3, we give the adiabatic potential energy curves (PECs), their vibrational level and their spectroscopic properties. In Section 3, we present our results of the transition and permanent dipole moments. Finally, Section 4 contains the conclusions.

2. Computational approach

2.1. Basis set

We have taken core polarizability of the francium atom from Aymar et al. (2006), which is α_Fr =20.38 $a_0^3.$ Then, ρ_s=3.1629 Bohr, ρ_p=3.027 Bohr and ρ_d=3.1068 Bohr are the optimized cutoff parameters. Whereas, to have an excellent representation of the atomic levels (7s; 7p; 6d; 8s; 8p; 7d; 9s and 9p), we have optimized a large Gaussian-type orbital (GTO) basis set which is 8s/6p/5d for the Fr atom. While for the cesium atom, we used this basis set (8s/6p/7d). Our choice of the basis set improves the calculated atomic level energies of the francium and cesium compared with previous studies, for example the work of Aymar et al. (2006).

In Table 1, we show our theoretical ionization energies of Fr and those for Aymar et al. (2006) which has been compared with the available experimental one (Ralchenko, Kramida, & Reader J NIST ASD Team, 2011). The examination of our theoretical and the experimental data indicates an excellent agreement between them (Ralchenko et al., 2011), and we observe that the biggest difference is 88 cm⁻¹. In addition, in Tables S1 and S2, we have been compared our molecular states beneath this ionic limit (Fr⁺+Cs^–) with the experimental ones for the molecule FrCs and the molecular ion (FrCs)⁺, respectively. The differences are acceptably small and it is in great accordance with the experimental data (Ralchenko et al., 2011). Clearly, in Table S1, the corresponding errors between the theoretical and experimental energies do not exceed 88 cm⁻¹.

Table 1. Theoretical ionization energies (in cm⁻¹) of francium atom compared with the experimental energies (Ralchenko et al., 2011).

Atomic levels	Expt. (Ralchenko et al., 2011)	This work	ΔE	Aymar et al. (2006)	ΔE
7s	–32848	–32848	0	–32 848	0
7p	–19487	–19487	0	–19 486	1
6d	–16499	–16499	0	–16 470	29
8 s	–13108	–13176	68	–13 161	53
8p	–9372	–9425	53	–9448	76
7d	–8551	–8463	88	–8394	157
9s	–7177	–7192	15	–7161	16
9p	–5565	–5580	15	–5619	54

ΔE: Energy difference between the experimental values and theoretical work in cm⁻¹.

Table 2. Spectroscopic constants for ^1,3Σ⁺, ^1,3Π and ^1,3Δ states of FrCs.

States	Re (a.u.)	De (cm^–1)	Te (cm^–1)	ωe (cm^–1)	ωeχe (cm^–1)	Be (cm^–1)×10^–5
(a) ^1Σ^+ States
X^1Σ^+	8.57	3517.97	0	36.859	0.0506	8.42
X^1Σ^+ [15]	8.55	3553	^–	^–	^–	^–
A^1Σ^+	9.98	4960.18	10905.69	26.118	0.0126	3.25
C^1Σ^+	10.81	3766.84	14186.84	22.113	0.0181	3.107
D^1Σ^+	11.26	1733.87	17402.14	15.995	0.0105	7.703
E^1Σ^+	10.90	2074.17	18558.77	16.637	0.0166	2.257
F^1Σ^+	9.55	3651.33	19074.54	33.869	0.3366	3.028
2nd min	18.67	2073.48	^–	^–	^–	^–
G^1Σ^+	10.13	2314.31	21528.27	15.403	0.0411	1.617
2nd min	18.42	1744.41	^–	^–	^–	^–
H^1Σ^+	9.56	3051.85	22730.99	19.694	0.666	8.901
2nd min	22.54	2898.92	^–	^–	^–	^–
I^1Σ^+	9.88	3186.11	23578.16	17.383	0.0322	2.016
2nd min	22.14	2442.99	^–	^–	^–	^–
3th min	35.75	722.35	^–	^–	^–	^–
J^1Σ^+	8.87	3328.96	24227.80	16.098	0.0304	1.959
2nd min	31.69	1443.89	^–	^–	^–	^–
3th min	45	665.66	^–	^–	^–	^–
K^1Σ^+	10.81	3974.37	24870.86	21.077	0.0305	2.401
2nd min	35.14	1582.84	^–	^–	^–	^–
3th min	49.61	904.82	^–	^–	^–	^–
L^1Σ^+	10.54	3860.06	25015.72	21.728	0.0997	2.263
2nd min	21.73	770.31	^–	^–	^–	^–
3th min	35.97	882.29	^–	^–	^–	^–
M^1Σ^+	10.012	2859.77	26707.87	18.612	0.0021	3.05
2nd min	43.71	524.61	^–	^–	^–	^–
(b) ^3Σ^+ States
a^3Σ^+	12.43	311.77	4762.59	12.31	0.0264	2.420
a^3Σ^+ [15]	12.33	217.9	^–	^–	^–	^–
c^3Σ^+	10.96	2148.09	14296.58	19.77	0.0046	7.411
d^3Σ^+	8.73	2014.59	14900.13	39.17	0.0342	3.864
e^3Σ^+	11.29	862.71	18130.79	11.231	0.0275	3.864
f^3Σ^+	12.59	1231.57	20979.58	11.3	0.032	3.76
g^3Σ^+	10.75	1665.49	21319.76	20.048	0.040	3.340
h^3Σ^+	10.51	2444.99	22946.07	29.661	0.027	3.82
i^3Σ^+	9.63	2928.34	23654.98	24.466	0.149	2.35
j^3Σ^+	9.83	2637.89	24423.14	27.029	0.161	1.25
k^3Σ^+	9.95	3178.47	24524.09	25.058	0.05	7.68
l^3Σ^+	10.06	3428.58	25083.75	19.837	0.055	1.053
m^3Σ^+	9.9	2703.02	25860.69	21.13	0.016	1.78
n^3Σ^+	10.61	2962.91	26167.96	17.454	0.0077	4.317
(c) ^1,3Π States
B^1Π	9.25	1468.1	13772.03	27.399	0.271	2.15
b^3Π	8.35	5979.77	9092.83	37.642	0.0269	4.42
1^1Π	9.97	1584.7	15887.77	20.754	0.0195	9.74
1^3Π	9.56	1547.95	15573.92	20.734	0.114	9.53
2^1Π	8.65	1031.84	17046.59	22.694	0.180	8.925
2^3Π	9.19	1556.1	16686.66	30.453	0.277	1.8
3^1Π	11.3	787.2	19976.58	22.694	0.18	8.925
3^3Π	11.62	1322.38	19833.92	17.601	0.0202	1.904
4^1Π	9.71	3973.07	21943.07	25.54	0.0299	1.38
4^3Π	9.69	4154.72	21774.08	23.34	0.117	1.148
5^1Π	9.86	3613.41	23172.13	22.79	0.002	1.759
5^3Π	9.91	3553.79	23246.75	13.28	0.034	2.26
6^1Π	9.56	3609.32	23769.10	28.105	0.088	7.213
6^3Π	9.54	3529.78	23863.48	27.612	0.211	1.92
7^1Π	10.4	3238.89	25397.60	17.8	0.049	6.93
7^3Π	7.09	1194.56	25312.01	23.27	0.086	7.68
8^1Π	7.76	1344.99	26380.85	21.25	0.0114	3.29
8^3Π	7.39	1606.46	26273.31	24.879	0.0192	3.843
(d) ^1,3Δ States
1^1Δ	8.57	2453.09	43537.18	29.984	0.0214	8.73
1^3Δ	8.56	2114.53	55088.13	29.257	0.118	8.994
2^1Δ	11.19	1031.46	57366.28	14.219	0.0041	1.729
2^3Δ	11.08	886.94	67152.65	15.633	0.00037	3.79
3^1Δ	9.45	3901.8	69547.12	29.339	0.105	7.27
3^3Δ	9.52	3867.26	72591.23	28.188	0.1139	1.22
4^1Δ	6.74	1215.11	73598.62	26.44	0.0255	7.511
4^3Δ	6.74	1216.52	78431.45	35.62	0.287	2.87

The present calculations display the ab initio approach proposed by Barthelat and Durand (1978) and Durand and Barthelat (1975) based on a nonempirical pseudo-potentials method to restrict the number of electrons to just two. Actually, our molecule is composed of 87 electrons for Fr and 55 electrons for Cs. The (FrCs)⁺ and FrCs are considered as an effective one and two valence electron, respectively. In our computation, the theoretical energies were determined at self-consistent field (SCF) level based on the core polarization potentials operator (CPP) method. Then, the full CI was calculated using the package code, which is developed by the LCPQ in Toulouse (CIPSI, MOYEN, BDAV) (Barthelat & Durand, 1978; Berriche & Gadea, 1995; Boutalib & Gadea, 1992; Boutalib, Daudey, & Mouhtadi, 1992; Chaieb, Habli, Mejrissi, Oujia, & Gadea, 2014; Dardouri, Habli, Oujia, & Gadéa, 2012; Duplaa & Spiegelmann, 1996; Durand & Barthelat, 1975; Evangelisti, Daudey, & Malrieu, 1983; Foucrault, Millie, & Daudey, 1992; Gadea & Pelissier, 1990; Groß & Spiegelmann, 1998; Habli, Dardouri, Oujia, & Gadéa, 2011; Habli et al., 2015, 2016; Hamdi et al., 2018; Huron, Malrieu, & Rancurel, 1973; Khémiri, Dardouri, Oujia, & Gadea, 2013; Khelifi, Oujia, & Gadea, 2007; Mtiri et al., 2017; Pélissier, Komiha, & Daudey, 1988; Poteau & Spiegelmann, 1995; Souissi et al., 2017a, 2017b). For the model of the interaction between the valence electrons with the polarizable Fr⁺ and Cs⁺ cores, the core polarization potential V_CPP giving by Müller, Flesch, and Meyer (1984) is given by (1) $$V_{\text{cpp}}=-\frac{1}{2}\sum _{\gamma }{\alpha }_{\gamma }{\overrightarrow{f}}_{\gamma }{\overrightarrow{f}}_{\gamma }$$(1)

In this formula, ${\overrightarrow{f}}_{\gamma }$ is the electrostatic field that is at center γ generated through the valence electrons and all the other centers’ cores and ${\alpha }_{\gamma }$ is the dipole polarizability of the core γ, which is written as (2) $$\overrightarrow{f_{\gamma }}=\sum _i\frac{{\overrightarrow{R}}_{\gamma i}}{R_{{\gamma }^{}i}^3}{F}_l(R_{\gamma i},{\rho }_{\gamma }^l)-\sum _{{\gamma }^{\prime }\ne \gamma }{Z}_c\frac{{\overrightarrow{R}}_{\gamma \prime \gamma }}{R_{{\gamma }^{\prime }\gamma }^3}$$(2) ${\overrightarrow{R}}_{\gamma i}$ and ${\overrightarrow{R}}_{{\gamma }^{\prime }\gamma }$ are the vectors of the core–electron and that for core–core, respectively. We have taken the l-dependent cut off form developed by Foucrault et al. (1992) the operator of the cut off $F_1(R_{\gamma i},{\rho }_{\gamma })$ is written as: (3) $$F\left(R_{\gamma i},{\rho }_{\gamma }\right)=\{\begin{array}{c}0;R_{\gamma i}<{\rho }_{\gamma }\\ 1;R_{\gamma i}>{\rho }_{\gamma }\end{array}$$(3)

The formulation below presents the cutoff radius (4) $$F(R_{{\gamma }^{}i},{\rho }_{\gamma })=\sum _{l=0}^{\infty }\sum _{m=-l}^{+1}{F}_l(R_{{\gamma }^{}i},{\rho }_{\gamma })|lm\gamma 〉〈lm\gamma |$$(4)

Whereas, the operator $|lm\gamma 〉〈lm\gamma |$ is the spherical harmonic in the center of the core γ.

Besides, to activate the spectroscopic works at the theoretical and experimental levels, then, to offer a sight for the feature of FrCs, the vibrational levels were fitted by the method of least-squares specified as follows (5) $$E_{\mathrm{v}}=V(R_{\mathrm{e}})+{\omega }_{\mathrm{e}}(v+\frac{1}{2})-{\omega }_{\mathrm{e}}{\chi }_{\mathrm{e}}{(v+\frac{1}{2})}^2$$(5)

3. Results and discussions

3.1. Adiabatic PECs and spectroscopic properties of FrCs and (FrCs)⁺ systems

The potential energy curves provide a structure for the assessment of the collision determinations of the inelastic and elastic phenomena. Furthermore, new spectroscopic results give details concerning the binding of systems. These spectroscopic constants have been calculated by using the technique of interpolation of the potential energies while, ω_e and ω_eχ_e have been investigated from a fit of a vibrational levels’ energies. In this work, the PECs have been determined for a large spectrum of internuclear distances between the Cs⁺ ion and the Fr atom from 5 to 100 a.u. Moreover, the PECs are shown in Figures 1–3, respectively, for the ¹Ʃ⁺, ³Ʃ⁺ and ²Ʃ⁺. The investigated spectroscopic parameters have been compared with the obtainable data and registered in Tables 2–4, respectively, for FrCs and (FrCs)⁺ systems.

Figure 1. Potential energy curves of the ¹Σ⁺ states of FrCs molecule.

Figure 2. Potential energy curves of the ³Σ⁺ states of FrCs molecule.

Figure 3. Potential energy curves of the ²Σ⁺ states of the molecular ion (FrCs)⁺.

Table 3. Spectroscopic constants for ²Σ⁺, ²Π and ²Δ states of (FrCs)⁺.

States	Re (a.u.)	De (cm^–1)	Te (cm^–1)	ωe (cm^–1)	ωeχe(cm^–1)	Be (cm^–1)×10^–8
(a) ^2Σ^+ States
X^2Σ^+	9.73	5048	0	30.36	0.035	3.479
X^2Σ^+ [15]	9.72	5052	^–	^–	^–	^–
A^2Σ^+	19.02	294.654	11586.07	7.39	0.054	9.066
A^2Σ^+ [15]	19.06	295.7	^–	^–	^–	^–
C^2Σ^+	17.05	3365.69	17893.77	12.4	0.006	1.133
D^2Σ^+	11	370.47	23341.13	4.3	0.009	2.722
2nd min	29.5	442.87	^–	^–	^–	0.378
E^2Σ^+	16	637.79	26479.61	5.6	0.009	1.28
F^2Σ^+	26	285.32	32273.74	4.76	0.002	0.487
G^2Σ^+	29.32	1236.9	35241.04	6.08	0.005	0.383
H^2Σ^+	42.94	198.55	39518.61	2.45	0.006	0.178
I^2Σ^+	19	557.5	40837.64	4.42	0.002	0.9124
2nd min	40.2	1492.73	^–	^–	^–	0.203
J^2Σ^+	27	147.05	41030.78	2.05	0.003	0.4518
2nd min	55.97	367.94	^–	^–	^–	0.105
K^2Σ^+	37.31	467.51	43271.62	4.34	0.002	0.236
L^2Σ^+	40	501.94	43971.74	2.35	0.001	0.2058
M^2Σ^+	54.58	408.9	46563.74	2.278	0.001	0.110
N^2Σ^+	24	159.56	47039.99	2.34	0.006	0.572
2nd min	74.66	44.35	^–	^–	^–	0.0590
O^2Σ^+	70.76	74.87	47375.79	1.18	0.007	0.06578
(b) ^2Π States
1^2Π	8.61	4380.84	13798.37	20.05	0.054	4.443
2^2Π	9.5	74.87	24085.15	3.23	0.025	3.64
2nd min	24.33	99.61	^–	^–	^–	0.556
3^2Π	19	504.35	24982.79	2.45	0.002	0.9124
4^2Π	22	485.04	30035.10	4.62	0.004	0.6805
5^2Π	28.27	1414.14	36079.43	5.51	0.002	0.412
6^2Π	35.46	184.03	39683.21	1.81	0.003	0.261
7^2Π	30	234	40155.08	2.41	0.004	0.365
8^2Π	42	104.7	44281.20	1.45	0.005	0.186
9^2Π	45.8	84.87	45097.65	1.12	0.006	0.157
(c) ^2Δ States
1^2Δ	7	87.9	21767.49	1.1	0.006	6.72
2^2Δ	18	245.7	25344.94	2.31	0.002	1.02
3^2Δ	25	356.86	36862.96	2.02	0.005	0.0527
4^2Δ	30	224.3	40644.51	1.95	0.004	0.36597

Table 4. Avoided crossing positions.

States	Positions (a.u.)	ΔE (cm^–1)
FrCs
E^1Σ^+/F^1Σ^+	9.2/19.5	348.97/546.49
F^1Σ^+/G^1Σ^+	14/26	1316.85/967.88
G^1Σ^+/H^1Σ^+	13/24/28	1187.36/54.87/15.36
H^1Σ^+/I^1Σ^+	6.6/12.5/39	144.85/531.13/280.93
I^1Σ^+/J^1Σ^+	7.1/9.8/32	695.73/590.39/19.75
J^1Σ^+/K^1Σ^+	7.3/11/36	114.13/144.85/50.48
K^1Σ^+/L^1Σ^+	7.8/11/22/66	19.75/21.95/81.21/13.19
L^1Σ^+/M^1Σ^+	18	568.44
c^3Σ^+/d^3Σ^+	8.2	414.81
f^3Σ^+/g^3Σ^+	7.1	32.92
g^3Σ^+/h^3Σ^+	11	390.66
h^3Σ^+/i^3Σ^+	9	179.97
i^3Σ^+/j^3Σ^+	18	318.24
j^3Σ^+/k^3Σ^+	6.9/8.4/23	131.68/74.62/553.08
k^3Σ^+/l^3Σ^+	14	120.71
(FrCs)^+
D^2Σ^+/E^2Σ^+	6.2/16.1	1073.23/136.07
F^2Σ^+/G^2Σ^+	12.2	542.10
I^2Σ^+/J^2Σ^+	10.3	175.58
J^2Σ^+/K^2Σ^+	15.8	621.11

Our dissociation energy for the ground state (FrCs) and (FrCs)⁺ (D_e=3517.97 cm⁻¹ and D_e = 5048 cm⁻¹) are in excellent accordance with the obtainable data found in the study of Aymar et al. (2006) (D_e =3553 cm⁻¹, D_e = 5054 cm⁻¹). For the excited states A¹Σ⁺, C¹Σ⁺, D¹Σ⁺ and E¹Σ⁺ of FrCs, we observe in Figure 1 curves with regular forms and have just one potential well located at equilibrium distances Re equal to (9.98 a.u, 10.81 a.u, 11.26 a.u and 10.90 a.u), respectively. Moreover, the profound of these wells were differ from 1700 to 5000 cm⁻¹ (see Table 2a). Besides, for the other excited states of FrCs, we can see curves with double and triple potential wells. Take an example, the F¹Σ⁺ state has two minimums where the first one (R_e =9.55 a.u. and D_e =3651.33 cm⁻¹) is deeper than the second one (R_e =18.67 a.u and D_e=2073.48 cm⁻¹). We can observe an important avoided crossing with the I¹Σ⁺ state observed at the distance R_AC=32 a.u, where the feeble energy difference between the two states is 20 cm⁻¹. Therefore, this minimal difference gives us adiabatic transitions. Moving on to the highest states, we observe many potential wells and the avoided crossings were specified. Then, for the molecular ion (FrCs)⁺, their curves in Figure 3 have single and double wells. For example, we can explain the state of D²Σ⁺, which has two wells, the first at 11 a.u. and the second at 29.5 a.u (see Table 3), depending on the avoided crossing with the adjacent E²Σ⁺ state at R_AC =16.1 a.u, where the small difference of energy in this position is relative to 136 cm⁻¹.

We have been displayed the curves of symmetry ³Σ⁺ of the molecule FrCs in Figure 2. We note that the first state a³Σ⁺ is almost an attractive state, where D_e =311.77 cm⁻¹ and R_e =12.43 a.u., which is in good accordance with the results found in the work of Aymar et al. (2006), D_e =217.9 cm⁻¹, R_e =12.33 a.u. (see Table 2b). Going to higher states, their shapes are involved with various potential wells: There is double, triple as well as multiple. We observed at short distances a set of avoided crossings between the examined states, which is leading to abnormal oscillation comportment in their potential energy curves. We have been displayed in Table 4 the difference of the energy between the states at the avoided crossings' positions PAC.

Turning now to the symmetries ^1,3Π and ²Π of the two systems FrCs and (FrCs)⁺, we observe that various states are repulsive in Figures 4 and 5 like the 2 ^1,3Π and 3 ²Π states. The adiabatic ^1,3Π curves of the molecule FrCs have nearly degenerated as exposed in Figure 4. Therefore, we can observe several avoided crossings, which a few of them are corresponding to underlying of the charge transfer states and the others are related to the interaction between attractive and repulsive curves. The whole states of Π symmetry singlet and triplet attained their asymptotic limits rapidly at R_e =40 a.u.

Figure 4. ^1,3Π adiabatic potential energy curves of the FrCs molecule.

Figure 5. ²Π adiabatic potential energy curves of the molecular ion (FrCs)⁺.

For the symmetries ^1,3Δ and ²Δ of FrCs and (FrCs)⁺, we can see in Figures 6 and 7 that all the curves are repulsive. In Tables 2d and 2c, we have been presented their spectroscopic parameters, respectively. We notice that these electronic states quite quickly achieve their asymptotic limits (exactly at R_e at about 30 a.u. for the two systems) and the ^1,3Δ curves with the identical dissociation limit are nearly degenerate. To recapitulate, the spectroscopic parameters for the triplet and singlet states achieved the uniform limits are nearly equal.

Figure 6. ^1,3Δ adiabatic potential energy curves of the FrCs molecule.

Figure 7. ²Δ adiabatic potential energy curves of the molecular ion (FrCs)⁺.

3.2. Vibrational levels of the molecule FrCs

The vibrational levels were investigated for the whole symmetries of FrCs. We start by the Figure 8, which is presented the PECs of the (X and F) ¹Σ⁺ with the vibrational levels spacing (G_v–G_v–1) related to the vibrational number of the levels “V”. For the ground state X¹Σ⁺, we observe a deep well that is contains 170 vibrational levels, which show a linear comportment considering an anharmonic shape as Morse in their potential energy curves. In addition, the spacing becomes tiny and vanish at this asymptotic limit (Fr(7s)+Cs(6s)). Similarly, we observe an identical form for the states, which have only one well depth. Moving on to the spacing for F¹Σ⁺, the behaviors present a shape that is linear up to v = 71 corresponding to an anharmonic potential at R_e =9.55 a.u. Then, we can see an abrupt variation, which indicates an appearance of the second depth at (18.67 a.u.), which is less deep than the other one. In Figure 9, we have been plotted the vibrational level spacing of (G–M) ¹Σ⁺. We note that for the highest states, their vibrational level spacing decreases in a linear behavior then an abrupt variation corresponded to the little enlargement of the first well. Therefore, these levels come to be greatly narrowed and we detect that quasi-degeneracy, which explain the appearance of the larger second well in the PECs.

Figure 8. Vibrational spacing (left) and potential energy curves (right) for (X and F) ¹Σ⁺ states of FrCs.

Figure 9. Vibrational level spacing for (G–M) ¹Σ⁺ states of FrCs.

Moving on to the vibrational levels spacing for ³Σ⁺ states, we can see for (a–e) ³Σ⁺ a linear shape related to the simple depth in their PECs (see Supplementary Materials, Figure S1). The higher states as k and m ³Σ⁺ states show an abrupt change in their spacings, which indicates an appearance of the second well (see Supplementary Materials, Figure S2).

3.3. Permanent and transition dipole moment of FrCs

In Figures 10 and 11, we have been reported the adiabatic PDM and TDM from all symmetries ^1,3Σ⁺. We start with Figure 10, where we can see the PDM functions of ¹Σ⁺ states differ leisurely at the short and large internuclear distances; they show an abrupt variation (see Figure 10b). Every state one after one; conduct to a maximum then drops to zero. Moreover, these adiabatic curves create a linear form with the fragments of the (–R) function as of the ionic state (Fr⁺+Cs^–) and move via creation knot between consecutive pieces when are combined. In addition, we note that the avoided crossings are very feeble because of the acuity of the slopes surrounding the crux of the dipole. The avoided crossings in the PECs and the crossings in the permanent dipole moment curves were caused by the production of the ionic curve. As consequences of these crossings that they are produced by the charge transfer or excitation efficacy. Such as, the neutralization cross-sections fundamentally depended on the crossing chains, thus, many astrophysical conditions (Barklem, Belyaev, Dickinson, & Gadea, 2010; Belyaev, Barklem, Dickinson, & Gadea, 2010; Croft, Dickinson, & Gadéa, 1999; Dickinson, Poteau, & Gadea, 1999) are their significant field for charge transfers. Indeed, the permanent dipole moment curves yields a direct figure of the ionic nature in ¹Σ⁺ of the electronic wave function.

Figure 10. (a) Permanent dipole moment for the ¹Σ⁺ states for the FrCs and (b) zoom for the PDM at short distance.

Figure 11. (a) Permanent dipole moment for the ³Σ⁺ states for the FrCs and (b) zoom for the PDM at short distance.

The PDM curves for ³Σ⁺, ^1,3Π and ^1,3Δ are displayed in Figures 11 and S3–S4. We can see in Figure 11 the PDM curves for ³Σ⁺ symmetry where the specified variations are presented at short distances before35 a.u. Furthermore, the PDM curves are very important for they have a significant influence on other excited states. In addition, they increase and disappear at large internuclear distances. For example, j³Σ⁺ and h³Σ⁺ states show a maximum (μ_max = −7.49 a.u. and −5.74 a.u.) respectively, at the distances R_e =21 a.u and 16.5 a.u., respectively. After that, it leisurely decreases. Moreover, the last maximums are related to the avoided crossings in their PEC_S. The permanent dipole moment figures of the symmetry ^1,3Π present a significant variation at short distances then drop quickly to zero (see Supplementary Materials, Figure S3). In the Supplementary Materials Figure S4, we can see that the permanent dipole moment functions of ¹Δ are uniform to the ³Δ ones that clarify the degeneracy in their potential energies curves.

To complete this work, it is important to illustrate the curves of the transition dipole moment in the adiabatic representation. In Figures 12 and 13, we have been presented the curves of ¹Σ⁺ and ³Σ⁺, which have been determined for the first time. In Figure 12, we can see the presence of several peaks, which are related to the avoided crossings in the potential energies curves at certain distances. For example, the F–G transition dipole moment that indicates a peak at about 8.1a.u. is related to the avoided crossing between the states F and G. Therefore, the high peak is observed for the H–I transition dipole moment figures, which are related to the crossing of the H–I the potential energies curves at the distance 13.5 a.u. To explain more, the avoided crossing was compared to the example of A–C when the peak is higher and less large. We have been displayed in Figure 13 the TDM figures for symmetry ³Σ⁺. Interestingly, the transition dipole moment figures have not a peak that is corresponding to the repulsive form in their potential energies curves and the absence of the avoided crossing. Further, the (c–d) ³Σ⁺ show a peak at R_e= 8.1 a.u., which is related to the P_AC between c and d states in their PECs.

Figure 12. Transition dipole moment for the ¹Σ⁺ states for the FrCs.

Figure 13. Transition dipole moment for the ³Σ⁺ states for the FrCs.

To sum up, the related avoided crossings in the PECs are observed around the wide internuclear distances and the corresponding maximum data in the transition dipole moment curves. The significant transitions which are between the adjacent states (i, i + 1), as, the transitions between (J–K) ¹Σ⁺ and (L–M) ¹Σ⁺ states are more significant than the others.

4. Conclusion

In this theoretical study, we have been determined the electronic structure of the two systems FrCs and (FrCs)⁺ in several symmetries ^1–3Σ⁺, ^1–3Π and ^1–3Δ. We have been studied the potential curves and their spectroscopic parameters by using the full configuration interaction (FCI) method. Comparing our results and the available results in the literature, we can see a good agreement between them. Furthermore, we have been determined the vibrational properties for the molecule FrCs by using a least-squares approach. Then, the PDM and TDM for the FrCs molecule have been determined for all symmetries.

Moreover, our new results of the permanent dipole moment functions present the presence of (–R) curve for the ionic limit (Fr⁺+Cs^–). In addition, the results show the several crossings in the potential energies curves data form the abnormal behaviors with numerous depths, which are corresponding to a linear shape in the PECs and the peaks in the transition dipole moment figures. To explain more, the potential energies curves create fragment by fragment for the (–R) function as a linear shape of the ionic state (Fr²⁺Cs^–) and move on creating a crux between consecutive pieces when combined. These crossings are a consequence of the radiative the photo-association or charge exchange and spontaneous emission (Croft et al., 1999; Dickinson et al., 1999). These new data are very significant and helpful for further experimental and theoretical researches, as in spectroscopy (Malta et al., 1997) or collision fields (Barklem and O'Mara 2002; Hudson, Gilfoy, Kotochigova, Sage, & De Mille, 2008; Hummon et al., 2011; Staanum, Kraft, Lange, Wester, & Weidemüller, 2006; Zahzam, Vogt, Mudrich, Comparat, & Pillet, 2006).

Disclosure statement

These new results are very important and helpful for further theoretical and experimental researches, for example in spectroscopy or collision fields.

Additional information

Funding

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through, the Fast-track Research Funding Program.