^{1}

^{2}

^{1}

^{*}

The theoretical investigation of the potential energy curves, in the representation
^{2s+1}Λ
^{(+/-)}, of the 27 low-lying Doublet and Quartet electronic states of the BP+ molecular ion has been performed with the methods in quantum chemistry, the Complete Active Space Self Consistent Field (CASSCF) and the Multireference Configuration Interaction (MRCI) calculations. The harmonic vibrational frequency
*ω*
_{e}, the inter-nuclear distance at equilibrium R
_{e}, the rotational constant B
_{e}, the electronic energy with respect to the minimum ground state energy T
_{e}, and the permanent dipole moment have also been calculated. Twenty-three new electronic states have been investigated here for the first time. The comparison between the values of the present work and those available in the literature for several electronic states shows a good agreement. These investigated data can be a conducive to further work on BP
^{+} molecular ion in both experimental and theoretical research.

The theoretical calculations and potential energy function of diatomic molecules have been one of several molecular physics important researches [^{−l} or, and 0 K was used as the reference temperature, but they did not give other experimental data on the BP radical. BP^{+} ion-molecule, attracted much attention as refractory semiconductor due to its physical and chemical properties [^{+} was X^{4}Σ^{−} and they determined their spectroscopic constants T_{e}, R_{e}, ω_{e}, ω_{e}χ_{e} and B_{e} for the first time by using MRCI calculations. The physical properties of BP^{+} took less attention than the BP molecule [^{+} ion-molecule acquired much focus. Multi-reference configuration interaction method was applied by Guo et al. [^{+} ground and the lowest 5 excited electronic states.

The present work is devoted to an accurate description of the ground and electronically 26 excited doublet and quartet electronic states. In order to emphasize the accuracy of our work we used the multi-reference configuration interaction (MRCI+Q) model expansion. The potential energy curves, the spectroscopic constants R_{e}, T_{e}, B_{e}, and ω_{e} for each of the corresponding electronic states have been investigated along with their static dipole moment.

The Hartree-Fock method relies on averages; it does not consider the instantaneous electrostatic interactions between electrons, nor does it take into account the quantum mechanical effects on electron distributions because the effect of n-1 electrons, on an electron of interest, is treated in an average way. This method ignores the electron correlation which is the difference in energy between the HF and the lowest possible energy.

A singly, doubly, triply, quadruply, …, N-tuply excited states is called a Configuration State Function (CSF). The exact ground and excited-states wavefuntions can be expressed as a linear combination of all possible N- electron Slater determinants arising from a complete set of spin orbitals

where all the C’s are expansion coefficients that must converge to zero and this method is called Configuration Interaction (CI). A simple formula known as the Davidson Correction has been proposed for estimating the contribution ΔE_{Q} of quadruply excited determinants to the correlation energy

The development of efficient multiconfiguration self-consistent field method MCSCF is particularly important for excited states. One such scheme is the Complete Active-Apace Self-Consistent Field method (CASSCF). In this approach, the spinorbitals are divided into three classes: 1) set of inactive orbitals composed of the lowest energy spinorbitals which are doubly occupied in all determinants, 2) a set of virtual orbitals of very high energy spinorbitals which are unoccupied in all determinants, 3) a set of active orbitals which are energetically intermediate between the inactive and virtual orbitals. In the conventional CI methods, the Hartree-Fock SCF wave function Ψ_{0 }is used as a reference configuration and the configuration state functions are formed by moving electrons out of the occupied spinorbitals of Ψ_{0} into unoccupied spinorbitals. In multireference configuration interaction (MRCI) a set of reference configurations is created from which excited determinants are formed for use in a CI calculation. Large scale MRCI wavefunctions can generate very accurate wavefunctions but are also computationally very intensive.

In the present work, the theoretical ab initio investigation of the low-lying doublet and quartet electronic states of the BP^{+} ion-molecule and its potential energy calculations for the states ^{2s+1}Λ^{(+/−)} have been performed via the CASSCF method [_{0}, 4s, 3d_{0}, 4p_{0}) and 4π (B: 2p_{±1}; P: 3p_{±1}, 3d_{±1}, 4p_{±1}) 1δ (P: 3d_{±2}) orbitals in the C_{2v} molecular orbitals are distributed into the irreducible representation a_{1}, b_{1}, b_{2} and 1a_{2} in the following way 6a_{1}, 4b_{1}, 4b_{2} , 1a_{2} noted [6, 4, 4, 1]. The 1s^{2} 2s^{2} of B and 1s^{2} 2s^{2} 2p^{6} 3s^{2} of P atoms were frozen in the MCSCF procedure. The number of active orbitals and valence electrons are 16 and 8 respectively.

The potential energy curves of the 26 doublet and quartet electronic states, in the representation ^{2s+1}Λ^{±} of the molecule BP^{+}, has been performed for 189 internuclear distances. These curves are given in Figures 1-4.

One can notice some avoided crossings between different doublet and quartet electronic states. The repulsion between the two atoms, when they are brought close to each other, is related to the Pauli principle: when the electronic clouds surrounding the atoms start to overlap, the energy of the system increases abruptly. Moreover, some crossings, of abscissa R_{c}, and avoided crossings, of abscissa R_{ac}, have been obtained between the potential energy curves of Figures 1-4. If the two curves correspond to states of different symmetry cross, they will ignore each other. Their wavefunctions Φ_{1} and Φ_{2} remain unperturbed and they are the adiabatic solutions of the Schrödinger equation. The crossing in this case is strictly allowed. But if these wavefunctions have the same symmetry, they will mix with each other to give two adiabatic solutions which no longer cross and the crossing

becomes avoided. The adiabatic solutions of the Schrödinger equation Ψ_{1} and Ψ_{2} are obtained by linear combinations of the diabatic ones where the variation method is used to solve for the coefficients. If H_{12} is the matrix element of the electronic hamiltonian H of the system between Φ_{1} and Φ_{2}, the energy gap E_{ac} between the two adiabatic states is twice the coupling term H_{12} at R_{ac}. The calculated values of R_{ac} and E_{ac} are given in

The spectroscopic constants such as the vibrational harmonic constant ω_{e}, the internuclear distance at equilibrium R_{e}, the rotational constant B_{e}, and the electronic transition energy with respect to the ground state T_{e} have been calculated by fitting the energy values around the equilibrium position to a polynomial in terms of the internuclear distance by using the data obtained from the ab initio calculation. These values are given in ^{4}Σ^{−}, (1)^{2}Σ^{−}, (1)^{2}Δ and (1)^{2}Π one can find that an excellent agreement is obtained for the values of R_{e} with the relative difference 0.5% < ΔR_{e}/R_{e} < 1.0% and a very good agreement for the values of B_{e} with the relative difference 1.1% < ΔB_{e}/B_{e} < 4.9% with a larger relative difference for the state (1)^{2}Π with the relative difference 8.2%. By comparing our calculated values of ω_{e} with those in literature we found a very good agreement with relative difference 0.3%< ΔB_{e}/B_{e} < 4.0% for the 3 states X^{4}Σ^{−}, (1)^{2}Σ^{−}, (1)^{2}Δ, while the agreement deteriorate for the state (1)^{2}Π with the relative difference 23%. For our values of T_{e} the relative difference is good for the (1)^{2}Σ^{−} with the relative difference 5.4% and becomes lager for the states (1)^{2}Δ and (1)^{2}Π with the relative differences 15% and 27% respectively. No comparison with other investigated states since, up to our knowledge, they are given here for the first time.

The permanent dipole moments are investigated for the considered electronic states of the ion-molecule BP^{+} by taking the boron atom at the origin, and the positions of the phosphorus atom are variable along the positive internuclear z-axis. All the calculations were performed with the MOLPRO [^{2}Σ^{−}, (2)^{2}Σ^{+}, (1)^{2}Δ, (1,2,3)^{2}Π, (1)^{4}Σ^{−}, (3)^{4}Π, tend to ± ∞ when R tends the ∞ which explains the ionic bounding of all these in these regions, 2) some other dipole moment curves, as (2)^{4}Σ^{−}, (1,2,4)^{4}Π, (1)^{4}Δ, (1)^{4}Σ^{+}, (3)^{4}Π, tend to constant limit where the bounding is covalent, 3) the agreement between the positions of the avoided crossings of the potential energy curves of 2 electronic states and the positions of the crossing of the dipole moment curves of these 2 states. This agreement may confirm the validity and the accuracy of the present calculation.

(n1) state 1/(n2) state 2 | R_{AC} (Å) | ΔE_{AC} (Hartree) |
---|---|---|

(2)^{2}Δ/(3)^{2}Δ | 2.59 | 0.0189 |

(2)^{4}Δ/(3)^{4}Δ | 2.17 | 0.0165 |

(2)^{2}Σ^{+}/(3)^{2}Σ^{+} | 2.65 | 0.0099 |

(2)^{4}Σ^{+}/(3)^{4}Σ^{+} | 2.38 | 0.0117 |

(1)^{2}Π/(2)^{2}Π | 2.20 | 0.0130 |

(2)^{2}Π/(3)^{2}Π | 1.69 | 0.0054 |

(2)^{4}Π/(3)^{4}Π | 2.44 | 0.0072 |

(3)^{4}Π/(4)^{4}Π | 2.23 | 0.0020 |

states | T_{e} (cm^{−}^{1}) | R_{e} (Ǻ) | ω_{e} (cm^{−}^{1}) | B_{e} (cm^{−}^{1}) |
---|---|---|---|---|

X^{4}Σ^{−} | 0.00^{a } 0.00^{b}^{ } 0.00^{c}^{ } | 1.863^{a } 1.872^{b } 1.861^{c} | 714.32^{a } 730.66^{b } 725.36^{c} | 0.6060^{a } 0.636^{b}^{ } 0.5991^{c} |

(1)^{2}Σ^{−} | 5317.89^{a } 4929.90^{b} | 1.883^{a } 1.881^{b} | 727.22^{a } 756.39^{b} | 0.6453^{a } 0.631^{b} |

(1)^{2}Δ | 8359.89^{a } 7105.21^{b} | 1.856^{a } 1.872^{b} | 791.18^{a } 781.82^{b} | 0.6107^{a } 0.636^{b} |

(2)^{2}Δ | 8451.92^{a} | 1.875^{a} | 110.38^{a} | 0.6267^{a} |

(1)^{2}Π | 8848.24^{a } 6452.89^{b} | 1.774^{a } 1.756^{b} | 812.31^{a } 1066.59^{b} | 0.6682^{a } 0.723^{b} |

(1)^{4}Π | 24,273.20^{a} | 1.785^{a} | 644.91^{a} | 0.656^{a} |

(1)^{4}Π | 30,017.27^{a} | 2.176^{a} | 451.06^{a} | 0.4419^{a} |

(2)^{4}Δ | 30,032.63^{a} | 2.160^{a} | 446.62^{a} | 0.4511^{a} |

(2)^{4}Σ^{−} | 30,040.59^{a} | 2.218^{a} | 228.81^{a} | 0.4465^{a} |

(2)^{2}Π | 30,681.12^{a} | 1.962^{a} | 602.47^{a} | 0.5459^{a} |

(1)^{4}Σ^{+} | 30,765.60^{a} | 2.198^{a} | 422.42^{a} | 0.4343^{a} |

(2)^{2}Σ^{−} | 31,382.01^{a} | 2.191^{a} | 535.44^{a} | 0.4382^{a} |

(2)^{4}Π | 34,121.08^{a} | 1.867^{a} | 980.77^{a} | 0.6034^{a} |

(3)^{2}Δ | 35,913.57^{a} | 2.264^{a} | 454.56^{ a} | 0.4101^{a} |

(2)^{2}Σ^{+} | 35,913.68^{a} | 2.234^{a} | 444.18^{a} | 0.4216^{a} |

(1)^{2}Φ | 37,485.07^{a} | 2.158^{a} | 418.16^{a} | 0.4516^{a} |

(4)^{2}Δ | 37,683.66^{a} | 1.609^{a} | 124.21^{a} | 0.8104^{a} |

(3)^{2}Σ^{+} | 40,530.78^{a} | 2.605^{a} | 544.38^{a} | 0.3099^{a} |

(5)^{2}Δ | 41,197.86^{a} | 2.654^{a} | 618.03^{a} | 0.2987^{a} |

(3)^{4}Π | 50,646.62^{a} | 2.075^{a} | 397.69^{a} | 0.4896^{a} |

(4)^{4}Π | 50,874.41^{a} | 2.282^{a} | 187.40^{a} | 0.4036^{a} |

(3)^{4}Σ^{−} | 52,564.41^{a} | 1.803^{a} | 288.00^{a} | 0.6473^{a} |

(2)^{4}Σ^{+} | 57,143.12^{a} | 1.772^{a} | 955.40^{a} | 0.6692^{a} |

(3)^{4}Δ | 61,768.15^{a} | 1.753^{a} | 928.38^{a} | 0.6824^{a} |

(4)^{4}Δ | 61,778.09^{a} | 1.784^{a} | 793.32^{a} | 0.6609^{a} |

(3)^{4}Σ^{+} | 61,824.45^{a} | 2.483^{a} | 472.28^{a} | 0.3406^{a} |

(5)^{4}Δ | 67,134.74^{a} | 2.253^{a} | 916.74^{a} | 0.4138^{a} |

^{a}Present Theoretical study using MRCI method; ^{b}Ref. [^{c}Ref. [

In the present work, ab initio calculation for 27 low-lying electronic states, in the representation ^{2s+1}Λ^{(+/−)}, of the BP^{+} ion-molecule were computed via CASSCF/MRSDCI methods for the electronic excited states. The potential energy curves, the electronic energy with respect to the ground state T_{e}, the harmonic frequency ω_{e}, the internuclear distance R_{e} have been calculated along with the rotational constant B_{e}. The comparisons of the present results with the available values in the literature show an overall very good agreement. To the best of our knowledge, 23 new electronic states have been investigated for the first time through this work. These new excited electronic states may provide a reliable theoretical basis and information for the experimental spectral properties related to the electronic structure for BP^{+} ion-molecule and the relevant sub-system [

MahdiMansour,NaylaEl-Kork,MahmoudKorek, (2015) Ab-Initio Calculations of 27 Electronic States of the BP^{+} Ion-Molecule. Journal of Modern Physics,06,1781-1788. doi: 10.4236/jmp.2015.613180