#
Symmetry Breaking of B_{2}N^{(−, 0, +)}: An Aspect of the Electric Potential and Atomic Charges

^{*}

## Abstract

**:**

_{2}N

^{(−, 0, +)}—radical, anion and cation—have been compared in terms of electric potential and atomic charges, ESP, rather than the well-known cut of the potential energy surface (PES). We have realized that the double minimum of the BNB radical is related to the lack of the correct permutational symmetry of the wave function and charge distribution. The symmetry breaking (SB) for B

_{2}N

^{(0, +)}exhibits energy barrier in the region of (5–150) cm

^{−1}. The SB barrier goes through a dynamic change with no centrosymmetric form which depends on the wave function or charge distribution. In spite of ${\tilde{A}}^{2}{\Sigma}_{g}^{+}$ exited state, the${\tilde{\text{}B}}^{2}{\prod}_{g}$ excited configuration contributes to the ground state (${\tilde{\text{}B}}^{2}{\prod}_{g}-{\tilde{X}}^{2}{\Sigma}_{u}^{+})$ for forming radicals. The SB did not occur for the anion form (B

_{2}N

^{(−)}) in any electrostatic potential and charges distribution. Finally, we have modified the Columbic term of the Schrödinger equation to define the parameters “αα' and ββ'” in order to investigate the SBs subject.

## 1. Introduction

_{∞h}” whereas its asymmetrically distorted form has the symmetry of the C

_{∞v}group. Tri-atomic B

_{2}N

^{(−, 0, +)}molecules have been subsequently studied using a variety of calculations and spectroscopic methods. It is a deep challenge to measure the real or artifactual SB effects due to its capability to a display a pseudo second-order Jahn–Teller effect, which results in a structure with unequal BN bond lengths [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

_{2}N has a symmetric linear regulation in its ground state (${\tilde{X}}^{2}{\Sigma}_{u}^{+})$) with an unusually low bending frequency (73 cm

^{−1}) [1].

^{(0)}–B is unstable with respect to symmetry lowering, i.e., the C

_{∞v}structure (r

_{1(BN)}≠ r

_{2(BN)}) yields a lower total energy than the D

_{∞h}symmetric structure (r

_{1(BN)}= r

_{2(BN)}) [10].

_{2}N

^{(0)}and cyclic B

_{2}N [4]. The UHF/6-31G* level of theory predicts the cyclic

^{2}B

_{2}state to be at the global minimum, while the correlated methods predict the ${\tilde{X}}^{2}{\Sigma}_{\left(u\right)}^{+}$ state of linear B–N–B to be at the global minimum and the cyclic

^{2}B

_{2}state to be at the local minimum.

_{∞v}{

^{2}Σ

^{+}:1σ

^{2}, 2σ

^{2}, 3σ

^{2}, 4σ

^{2}, 5σ

^{2}, 1π

^{4}, 6σ

^{2}, 7σ

^{1}}, and the second is the resonance form where the symmetry is D

_{∞h}{${(}^{2}{\Sigma}_{u}^{+}):1{\mathsf{\sigma}}_{g}^{2},1{\mathsf{\sigma}}_{u}^{2},2{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{u}^{2},1{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{1}$}. Obviously describing the electronic structure of the open-shell system (B

_{2}N

^{(0)}) is much more difficult than the closed-shell system of B

_{2}N

^{(−, +)}and in many cases, the description of an open-shell system is challenging in computational quantum chemistry calculations. This is primarily due to the presence of static correlation effects (requiring a multireference-type description) [13,14].

^{(0)}B has been examined via the developed reduced multireference coupled cluster method with singles and doubles that is perturbatively corrected for triples [RMR CCSD (T)] using the correlation consistent basis sets (cc-pVDZ, cc-pVTZ and cc-pVQZ) by J. Paldus [9]. They showed that the ground state has an asymmetric structure C

_{∞v}with two BN bonds of unequal length.

^{−1}above the ground state of the ${\tilde{X}}^{2}{\Sigma}_{\left(u\right)}^{+}$ [2]. He showed that the observed signal in the 355 and 266 nm photoelectron spectra of B

_{2}N

^{−}has been indicated to a photodetachment from the anion ground state ${\tilde{(X}}^{1}{\Sigma}_{g}^{+})$ to the ground and lowest excited states of neutral B

_{2}N i.e., ${\tilde{X}}^{2}{\Sigma}_{u}^{+}\text{}$ and $\text{}{\tilde{A}}^{2}{\Sigma}_{g}^{+}\text{}$ with a linear symmetry and is assigned to the ${\tilde{X}}^{1}{\Sigma}_{g}^{+}\text{}\to {\tilde{X}}^{2}{\Sigma}_{u}^{+}+\text{}{e}^{-}$ and ${\tilde{X}}^{1}{\Sigma}_{g}^{+}\text{}\to {\tilde{A}}^{\text{}2}{\Sigma}_{g}^{+}+\text{}{e}^{-}$ transitions.

_{2}N

^{(−)}anion [2] confirmed a linear symmetric geometry. Furthermore, the infrared absorptions were also observed in the cryogenic argon matrix near 6000 cm

^{−1}of the electronic band system, due to the ${\tilde{A\text{}}}^{2}{\Sigma}_{g}^{+}-{\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ [7].

_{2}N

^{(0)}radical [20]. Their results show that B

_{2}N in its ground state has a linear non-centrosymmetric structure with two equivalent global minima of the adiabatic potential energy surface, including two oppositely directed dipole moments, respectively. They accepted that the PJT effect involving vibronic interaction with the first excited state ${\tilde{A}}^{2}{\Sigma}_{\left(g\right)}^{+}$ via the asymmetric stretching vibrations is the major reason for the double-minimum. On the other hand, the large-scale multi-reference configuration interaction calculations, CASSCF+1+2 predicted an asymmetric configuration, while the SACASSCF+1+2 predicted a symmetric D

_{∞h}ground state [21].

^{−})/[3s2p] calculations and found no trace of SB in qualitative disagreement with all previous theoretical investigations that predicted a barrier to a centro-symmetric structure either of 20 cm

^{−1}(based on MRCI methods of Boggs) or of 100–160 cm

^{−1}(based on CC methods) [9,22,23].

_{2}N

^{(−, 0, +)}(anion, radical and cation) in terms of electrostatic potential charges “ESP” rather than the cut of the potential energy surface (ESP is changed using the trial wave functions). For a charged system with charge Q, the density |$\mathsf{\psi}\left(x\right){|}^{2}$ multiplied by the atomic charge yields the charge density |$Q\left(x\right){|}^{2}$. Large points for fitting of various situations have been used to calculate the atomic charges and electrostatic potential of the systems. As a result, the possibility of an asymmetric ground state may not be eliminated or it seems that an asymmetric geometry is a rather comfortable situation; the double minimum nature of BNB is related to the lack of the correct permutation symmetry of the wave functions (Scheme 1).

**Scheme 1.**A simplified of spontaneous symmetry breaking, (a) indicates high energy level which the ball settles in the center, and the result is symmetrical; (b) and (c) are lower in energy levels and the overall “rules” remain symmetrical, however, as the potential comes into effect, the “local symmetry” is inevitably broken since eventually the ball must roll one way (at random) and not another; (

**A**) is a non-linear combination of two wave function and (

**B**) is a non-linear combination of three wave function.

_{2}N

^{(−, 0, +)}systems have been estimated using symmetrical linear combination of wave functions (SLC-WFs). These barriers are dependent on the charge distribution, SLC-WFs and correlation effects between $|\mathsf{\alpha}\rangle \text{}$or$\text{}|\mathsf{\beta}\rangle $ levels which are not global minima.

_{2}N

^{(−, 0, +)}systems. Therefore, based on Walsh prediction [24], it appears that the linear p-block molecule holds up quite well for all B

_{2}N

^{(−, 0, +)}ions and the radical. This work has focused on a spontaneous symmetry breaking (SSB) [25,26,27] for B

_{2}N

^{(−, 0, +)}systems in view of ESP.

## 2. Wave Functions and Symmetry Breaking

_{2}N

^{(−, 0, +)}forms will be stable by minimization of “V”. So, it depends on sum of Coulomb repulsions between nuclei and the electronic effective potential:

_{N}, ${q}_{{B}_{1}},{q}_{{B}_{2}}$ are bound distances and charge distribution of B

_{2}N

^{(−, 0, +)}and ${\overrightarrow{\mathsf{\lambda}}}_{1}$, ${\overrightarrow{\mathsf{\lambda}}}_{2}$ and ${\overrightarrow{\mathsf{\lambda}}}_{3}$ are defined as:

_{N}× q

_{B}multiplication can be positive or negative and consequently, ⎢${\overrightarrow{\mathsf{\lambda}}}_{0}\u23a2=\sqrt[3]{\frac{{q}_{N}^{\left(0\right)}{q}_{B}^{\left(0\right)}}{8{\mathsf{\pi}\mathsf{\epsilon}}_{0}\mathsf{\alpha}{\mathsf{\alpha}}^{\prime}}}$ would be either positive or negative.

_{N}<+0.87 (for the anion form) and in equilibrium the charges of the nitrogen and the two borons are q

_{N}= 0.868802 and ${q}_{{B}_{1}}={q}_{{B}_{2}}={q}_{B}=-0.934401$. Therefore, the ${\overrightarrow{\mathsf{\lambda}}}_{0}=$ 1.329 and αα' = ββ' = 1.506 can be yielded in the center.

_{2}N

^{(−, 0, +)}radical, cation and anion forms, the charges of atoms always localize in a definite position in space. In fact, for a charged quantum system, it has been described by the wave function. Thus, the charges distribution with a certain amount in space and different distributions between borons and nitrogen atoms are important for the understanding of real or artifactual SBs problems of radical and ion BNB forms.

## 3. Computational Details

_{2}N

^{(0)}and 10 and 12 electrons for B

_{2}N

^{(+)}and B

_{2}N

^{(−)}respectively. In some part of our discussion, the BNB has been optimized via various levels of theory such as CASSCF (11, 12)/cc-pvqz and CASSCF (11, 12)/AUG-cc-pvqz (for radical) and CASSCF (10, 12)/cc-pvqz for cation. Approximation spin orbit coupling between two spin states has been computed during CASSCF calculations [45,46].

## 4. Results and Discussion

_{2V}/C

_{∞V}(global minima of ${\tilde{X\text{}}}^{2}$Σ

^{+}), are $E\left(|\mathsf{\alpha}\rangle \right)=-34.87083$ and $E\left(|\mathsf{\beta}\rangle \right)=-34.15046$ Hartree respectively, while these energies for C

_{∞V}/D

_{∞h}(local minima) are $E\left(|\mathsf{\alpha}\rangle \right)=-34.87079$ and $E\left(|\mathsf{\beta}\rangle \right)=-34.15042$, respectively. Although the energy of ${\tilde{A}}^{2}{\Sigma}_{\left(g\right)}^{+}$ state is near the ground state, this excited configuration does not contribute to the ground state wave function.

**Figure 1.**Relative energies of B

_{2}N

^{(−, 0, +)}versus B-N-B bond distance in various level of methods (

**a, a'**) cation; (

**b, b'**) radical and (

**c**) anion.

State | ${E}_{e}(Hartree)$ | $|\mathsf{\alpha}\rangle $Configuration | $|\mathsf{\beta}\rangle $Configuration | ${r}_{e}\left({B}_{1}N\right)$ | ${A}_{1}(2,1,3,-2,-1)$ |
---|---|---|---|---|---|

(*N_{e}) | Total Energy of $|\mathsf{\alpha}*\rangle $ | Total Energy of $|\mathsf{\beta}\rangle *$, Virtual ** | ${r}_{e}\left(N{B}_{2}\right)$ | ${A}_{2}(2,1,3,-1,-2)$ | |

$\left(Homo-Lumo\right)\text{}$ ** | |||||

$\tilde{X\text{}}$ ^{2}Σ^{+} | $-104.0781959{\text{}}^{a}$ | $\left[A\right],{\mathsf{\pi}}^{4},{\mathsf{\sigma}}^{2},{\mathsf{\sigma}}^{1}|\mathsf{\alpha}\rangle =-{0.44641}^{\text{}a}$ | $\left[B\right]{\mathsf{\sigma}}^{1}|\mathsf{\beta}\rangle =-0.26180{\text{}}^{a}$ | ${1.3189}^{\text{}a}$ | ${A}_{1}=179.9499{\text{}}^{a}$ |

$({C}_{\infty v})$ | $-{104.0820328}^{\text{}{a}^{\prime}}$ | $*E\left(|\mathsf{\alpha}\rangle \right)(-34.87083){\text{}}^{a}$ | $*{(-34.15046)}^{\text{}a}$ | $1.3185{\text{}}^{a}$ | ${A}_{2}=179.9602{\text{}}^{a}$ |

(*17e) | $-104.0754917{\text{}}^{{a}^{\prime \prime}}$ | $**{(-0.48614)}^{\text{}a}$ | $**(-0.17697){\text{}}^{a}$ | ||

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-104.0781959{\text{}}^{b}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{1}|\mathsf{\alpha}\rangle =-0.44641{\text{}}^{b}$ | $\left[{B}^{\prime}\right],4{\mathsf{\sigma}}_{g}^{1}|\mathsf{\beta}\rangle =-0.26180{\text{}}^{b}$ | $1.3187{\text{}}^{b}$ | ${A}_{1}=180.0{\text{}}^{b}$ |

$({D}_{\infty h})$ | $*(-34.87079){\text{}}^{b}$ | $*(-34.15042){\text{}}^{b}$ | ${1.3187}^{\text{}b}$ | ${A}_{2}=180.0{\text{}}^{b}$ | |

$**(-0.48614){\text{}}^{b}$ | $**(-0.17697){\text{}}^{b}$ | ||||

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-{103.6396773}^{\text{}d}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{1}|\mathsf{\alpha}\rangle =-\text{}0.42477{\text{}}^{d}$ | $\left[{B}^{\prime}\right],4{\mathsf{\sigma}}_{g}^{1}|\mathsf{\beta}\rangle $=-0.24844 ^{d} | $1.317$6 ^{d} | ${A}_{1}={180.0}^{\text{}d}$ |

$({D}_{\infty h})$ | $*(-34.74888){\text{}}^{d}$ | $*(-34.06924){\text{}}^{d}$ | $1.3176{\text{}}^{d}$ | ${A}_{2}=180.0{\text{}}^{d}$ | |

(*17e) | $**(-0.49245){\text{}}^{d}$ | ** $\text{}{\left(-0.18742\right)}^{d}$ | |||

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-104.159145{\text{}}^{f}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{1}|\mathsf{\alpha}\rangle =\text{}-{0.44701}^{\text{}f}$ | $\left[{B}^{\prime}\right],4{\mathsf{\sigma}}_{g}^{1}|\mathsf{\beta}\rangle =-0.26341$ | $1.3275{\text{}}^{f}$ | ${A}_{1}=180.0{\text{}}^{f}$ |

$({D}_{\infty h})$ | $-{104.1355512}^{\text{}n}$ | $*(-34.86501)$ | $*(-34.14329)$ | $1.3275$ | ${A}_{2}=180.0$ |

$**(-0.48249)$ | $**(-0.17779)$ | ||||

${\tilde{A\text{}}}^{2}{\Sigma}_{g}^{+}$ | $-104.1047582{\text{}}^{k}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{1},3{\mathsf{\sigma}}_{u}^{2}|\mathsf{\alpha}\rangle =-{0.44638}^{\text{}u}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},\text{}4{\mathsf{\sigma}}_{u}^{1},=-0.26134$ | $1.3154{\text{}}^{k}$ | ${A}_{1}=180.0{\text{}}^{k}$ |

(*17e) | $-{104.0781729}^{\text{}{K}^{\prime}}$ | $*(-34.8744){\text{}}^{u}$ | $*(-34.15408)$ | $1.3154{\text{}}^{k}$ | ${A}_{2}=180.0{\text{}}^{k}$ |

$**(-0.48626){\text{}}^{u}$ | $**(-0.17666)$ | ||||

${\tilde{\text{}B}}^{4}{\prod}_{g}$ | $-{104.0297022}^{\text{}h}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},{\mathsf{\pi}}_{g,}^{2}{\mathsf{\sigma}}_{g}^{1}|\mathsf{\alpha}\rangle =-0.26899{\text{}}^{h}$ | $\left[{A}^{\prime}\right],{\mathsf{\pi}}_{u}^{2}|\mathsf{\beta}\rangle -0.49646$ | $1.3079{\text{}}^{h}$ | ${A}_{1}=180.0{\text{}}^{h}$ |

(*17e) | $-104.0141196{\text{}}^{a}$ | $*(-35.04484){\text{}}^{h}$ | $*(-33.74801)$ | ${1.3079}^{\text{}h}$ | ${A}_{2}=180.0{\text{}}^{h}$ |

$*(-0.30125){\text{}}^{h}$ | $**(-0.50595)$ | ||||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | $-104.1965676{\text{}}^{a}$ | $\left[{A}^{\prime}\right]1{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{2}=-0.13904{\text{}}^{a}$ | $1.3291{\text{}}^{a}$ | ${A}_{1}={179.9322}^{\text{}a}$ | |

(*18e) | $-104.2019114{\text{}}^{{a}^{\prime}}$ | $*E\left(|\mathsf{\alpha}\rangle \right)=-32.48647{\text{}}^{a}$ | $1.3291$ | ${A}_{2}=179.9462$ | |

$-{104.1950169}^{\text{}{a}^{\prime \prime}}$ | $**-0.36286$ | ||||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | $-104.1965676{\text{}}^{b}$ | $*E\left(|\mathsf{\alpha}\rangle \right)=-32.48644{\text{}}^{b}$ | $1.3291{\text{}}^{b}$ | ${A}_{1}=180.0{\text{}}^{b}$ | |

(*18e) | $1.3291$ | ${A}_{2}=180.0$ | |||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | $-104.1145486{\text{}}^{c}$ | $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},{\mathsf{\sigma}}_{g}^{2},{\mathsf{\sigma}}_{u}^{2}=-{0.13454}^{\text{}c}$ | $1.3459{\text{}}^{c}$ | ${A}_{1}=179.8967{\text{}}^{c}$ | |

(*18e) | $-{104.1162881}^{\text{}{c}^{\prime}}$ | $1.3459{\text{}}^{c}$ | ${A}_{2}=179.9181{\text{}}^{c}$ | ||

$-104.112568{\text{}}^{{c}^{\prime \prime}}$ | |||||

${\tilde{\text{}A}}^{3}{\Pi}_{u}$ | $-104.0884685{\text{}}^{a}$ | [C]${\mathsf{\pi}}_{u}^{4},3{\mathsf{\sigma}}_{u}^{1},1{\mathsf{\pi}}_{g}^{1}|\mathsf{\alpha}\rangle $=$-0.03986{\text{}}^{h}$ | $[{C}^{\prime}]{\mathsf{\pi}}_{u,}^{4},4{\mathsf{\sigma}}_{g}^{1}|\mathsf{\beta}\rangle $ = $-0.01559{\text{}}^{h})$ | ${1.3422}^{\text{}h}$ | ${A}_{1}=180.0{\text{}}^{h}$ |

(*18e) | $-104.1164696{\text{}}^{h}$ | $*E\left(|\mathsf{\alpha}\rangle \right)=-{32.79721}^{\text{}h}$ | $*E\left(|\mathsf{\beta}\rangle \right)=-32.26335{\text{}}^{h}$ | $1.3422$ | ${A}_{2}=180.0$ |

$-104.0809447{\text{}}^{{a}^{\prime}}$ | $**-{0.25941}^{\text{}h}$ | $**-{0.14777}^{\text{}h}$ | |||

$-{104.0792592}^{\text{}{a}^{\prime \prime}}$ | |||||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | $-103.7454365{\text{}}^{a}$ | $[D](1{\mathsf{\pi}}_{u}^{4}),(4{\mathsf{\sigma}}_{g}^{2}$=$-0.57788{\text{}}^{a})$ | $1.2938{\text{}}^{a}$ | ${A}_{1}={180.7911}^{\text{}a}$ | |

(*16e) | $-103.8054934{\text{}}^{{a}^{\prime}}$ | $*E\left(|\mathsf{\alpha}\rangle \right)=-36.52419{\text{}}^{a}$ | $1.2938{\text{}}^{a}$ | ${A}_{2}=180.6276{\text{}}^{a}$ | |

$-103.7545006{\text{}}^{{a}^{\prime \prime}}$ | $**-{0.17926}^{a}$ | ||||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | $-103.6023122{\text{}}^{m}$ | $[D](1{\mathsf{\pi}}_{u}^{4}),(4{\mathsf{\sigma}}_{g}^{2}$=$-0.57803{\text{}}^{g})$ | ${1.3156}^{\text{}m\text{}}$ | ${A}_{1}={179.981}^{m}$ | |

(*16e) | $-103.3012055{\text{}}^{g}$ | $*E\left(|\mathsf{\alpha}\rangle \right)=-36.51985{\text{}}^{g}$ | $1.3156{\text{}}^{m}$ | ${A}_{2}={179.985}^{m}$ | |

$-103.8377401{\text{}}^{f}$ | $**-0.17944{\text{}}^{g}$ | $1.3003{\text{}}^{h}$ | |||

$-{103.7903312}^{\text{}h}$ | ${1.3004}^{\text{}h}$ | ||||

$\tilde{B\text{}}$ ^{3}Σ_{g} | $-103.7609202{\text{}}^{a}$ | $\{\left[{A}^{\prime}\right]4{\mathsf{\sigma}}_{g}^{1},3{\mathsf{\sigma}}_{u}^{1},{\mathsf{\pi}}_{u}^{4},\}{\text{}}^{a}$ | $\{\left[{A}^{\prime}\right]\text{},{\mathsf{\pi}}_{u}^{2}|\mathsf{\beta}\rangle =-0.75222\}{\text{}}^{a}$ | $1.2976{\text{}}^{a}$ | ${A}_{1}={180.0}^{\text{}a}$ |

(*16e) | $-103.7767141{\text{}}^{h}$ | ${\mathsf{\pi}}_{u}^{2}|\mathsf{\alpha}\rangle =-0.74323{\text{}}^{a}$ | $*E\left(|\mathsf{\beta}\rangle \right){(-35.72475)}^{a\text{}}$ | ${1.2976}^{\text{}a}$ | ${A}_{2}=180.0$ |

$-103.7628505{\text{}}^{{a}^{\prime}}$ | $*E\left(|\mathsf{\alpha}\rangle \right)(-37.33372){\text{}}^{a}$ | $**(-0.52131){\text{}}^{a}$ | |||

$-103.7590557{\text{}}^{{a}^{\prime \prime}}$ | $**(-0.53385){\text{}}^{a}$ |

^{(a)}QCISD/EPR-II,

^{(a'')}MP

_{4}D/EPR-III//QCISD/EPR-III,

^{(m)}CASSCF

^{(a'')}MP

_{4}SDQ/EPR-III//QCISD/EPR-III,

^{(d)}CASSCF (11, 12)/UHF,

^{(g)}CASSCF(10,12)rohf AUG-cc-pvqz,

^{(b)}QCISD/EPR-III(${\mathsf{\theta}}_{const}=180.0)$, $\left[\text{}A\text{}\right]:1{\mathsf{\sigma}}^{2},2{\mathsf{\sigma}}^{2},3{\mathsf{\sigma}}^{2},4{\mathsf{\sigma}}^{2},5{\mathsf{\sigma}}^{2}$,

^{(c)}QCISD/EPR-II, $\left[{A}^{\prime}\right]:1{\mathsf{\sigma}}_{g}^{2},1{\mathsf{\sigma}}_{u}^{2},2{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{u}^{2}$;

^{(c')}MP

_{4}D/EPR-II//QCISD/EPR-II, $\left[B\text{}\right]:\text{}{\mathsf{\sigma}}^{1},{\mathsf{\sigma}}^{1},{\mathsf{\sigma}}^{1},{\mathsf{\sigma}}^{1},{\mathsf{\sigma}}^{1},{\mathsf{\pi}}^{2}$;

^{(c'')}MP

_{4}SDQ/EPR-II//QCISD/EPR-II, [${B}^{\prime}$]: $1{\mathsf{\sigma}}_{g}^{1},2{\mathsf{\sigma}}_{g}^{1},1{\mathsf{\sigma}}_{u}^{1},3{\mathsf{\sigma}}_{g}^{1},2{\mathsf{\sigma}}_{u}^{1},1{\mathsf{\pi}}_{u}^{2}$,

^{(f)}CBS-lq,

^{(h)}QCISD(T)/EPR-III, [C]: $1{\mathsf{\sigma}}_{g}^{2},1{\mathsf{\sigma}}_{u}^{2},2{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{u}^{2},4{\mathsf{\sigma}}_{g}^{2}$, $\left[{C}^{\prime}\right]:1{\mathsf{\sigma}}_{g},1{\mathsf{\sigma}}_{u},2{\mathsf{\sigma}}_{g},3{\mathsf{\sigma}}_{g},2{\mathsf{\sigma}}_{u}$,

^{(n)}CBS4O,

^{(u)}TD/EPR-II

^{(k)}TD/EPR-III//QCISD (T)/EPR-III,

^{(k')}TD/EPR-III//QCISD /EPR-III, [D]: $1{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{g,}^{2}1{\mathsf{\sigma}}_{u}^{2},3{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{u}^{2}$. Total energy of $|\mathsf{\beta}\rangle $ and the energy of virtual orbital are shown with “*” and “**” symbols, respectively.

**Table 2.**NBO, electric potential, gradient of electric potential $(\mathcal{L})$, atomic occupancies, and Fock Matrix data of B

_{2}N

^{(−, 0, +)}in ground and exited states.

State | ${B}^{E{p}_{1}}\u2013{N}^{E{p}_{2}}\u2013{B}^{E{p}_{3}}$ | $\mathcal{L}=\frac{{E}_{P{(}_{B})}-{E}_{P{(}_{N})}}{R(BN)}$ | Hybrids Coefficient& | ${E}_{accepto{r}_{(j)}}-{E}_{Dono{r}_{(i)}}$ | Atomic Occupancies |
---|---|---|---|---|---|

(*N_{e}) | $*Fock\text{}Matrix\text{}({F}_{i,j},a.u.)$ | ||||

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | ${N}^{E{p}_{2}}=-18.3638{\text{}}^{a}$ | $(5.32)$, $\text{}(5.32)$ | ${|\mathsf{\psi}\rangle}_{BD\left(1\right)}=0.91S{P}_{{N}_{1}}^{1.01}+0.41\text{}S{P}_{{B}_{2}}^{{2.46}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(2\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=2.01$ | $|\mathsf{\alpha}\rangle N:2{s}^{0.77}2{P}_{x}^{0.84}2{P}_{y}^{0.84}2{P}_{z}^{0.87}$ |

(*17e) | ${B}^{E{p}_{1}}={B}^{E{p}_{3}}-{11.3464}^{\text{}a}$ | ${|\mathsf{\psi}\rangle}_{BD(2)}=0.96S{P}_{{N}_{1}}^{1.0}+0.29S{P}_{{B}_{2}}^{{1.0}^{a}}$ | * 0.094 | $|\mathsf{\alpha}\rangle B:2{s}^{0.76}2{P}_{x}^{0.79}2{P}_{y}^{0.79}2{P}_{z}^{0.40}$ | |

${|\mathsf{\psi}\rangle}_{BD(3)}=0.96S{P}_{{N}_{1}}^{1.0}+0.29S{P}_{{B}_{2}}^{{1.0}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=1.00$ | $|\mathsf{\beta}\rangle N:2{s}^{0.80}2{P}_{x}^{0.86}2{P}_{y}^{0.86}2{P}_{z}^{0.83}$ | |||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}=0.28S{P}_{{N}_{1}}^{1.0}-0.95\text{}S{P}_{{B}_{2}}^{{1.0}^{a}}$ | * 0.089 | $|\mathsf{\beta}\rangle B:2{s}^{0.50}2{P}_{x}^{0.69}2{P}_{y}^{0.69}2{P}_{z}^{0.16}$ | |||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(2\right)}=0.42S{P}_{{N}_{1}}^{1.01}-0.90\text{}S{P}_{{B}_{3}}^{{0.95}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}-{|\mathsf{\psi}\rangle}_{BD\left(2\right)}=0.7$ | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}=0.70S{P}_{{B}_{2}}^{0.97}-0.70\text{}S{P}_{{B}_{3}}^{{0.97}^{a}}$ | * 0.029 | ||||

${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ | ${N}^{E{p}_{2}}=-{18.6013}^{\text{}a}$ | $(5.25)$, $\text{}(5.25)$ | ${|\mathsf{\psi}\rangle}_{BD\left(1\right)}=0.90S{P}_{{N}_{1}}^{1.01}+0.43\text{}S{P}_{{B}_{2}}^{{2.26}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=1.94$ | $|\mathsf{\alpha}\rangle ,\mathsf{\beta}\rangle N:2{s}^{1.59}2{P}_{x}^{1.70}2{P}_{y}^{1.70}2{P}_{z}^{1.66}$ |

(*18e) | ${B}^{E{p}_{1}}={B}^{E{p}_{3}}-{11.6190}^{\text{}a}$ | ${|\mathsf{\psi}\rangle}_{BD(2)}=0.66S{P}_{{N}_{1}}^{1.0}+0.75S{P}_{{B}_{2}}^{{1.0}^{a}}$ | * 0.107 | $|\mathsf{\alpha}\rangle ,\mathsf{\beta}\rangle B:2{s}^{0.97}2{P}_{x}^{0.14}2{P}_{y}^{0.14}2{P}_{z}^{0.37}$ | |

${|\mathsf{\psi}\rangle}_{BD(3)}=0.95S{P}_{{N}_{1}}^{1.0}+0.32S{P}_{{B}_{2}}^{{1.0}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=0.73$ | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}=0.44S{P}_{{N}_{1}}^{1.01}-0.90\text{}S{P}_{{B}_{2}}^{{2.24}^{a}}$ | *0.021 | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(2\right)}=0.75S{P}_{{N}_{1}}^{1.0}-0.67\text{}S{P}_{{B}_{3}}^{{1.0}^{a}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}-{|\mathsf{\psi}\rangle}_{BD\left(3\right)}=1.94$ | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}=0.32S{P}_{{N}_{1}}^{1.0}-0.94\text{}S{P}_{{B}_{3}}^{{1.0}^{a}}$ | * 0.107 | ||||

$\tilde{B\text{}}$ ^{3}Σ^{+} | ${N}^{E{p}_{2}}=-18.10657{\text{}}^{h}$ | $(5.43)$, $\text{}(5.43)$ | ${|\mathsf{\psi}\rangle}_{BD\left(1\right)}=0.90S{P}_{{N}_{1}}^{1.0}+0.42\text{}S{P}_{B}^{{0.72}^{\text{}h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=1.98$ | $|\mathsf{\alpha}\rangle N:2{s}^{0.78}2{P}_{x}^{0.81}2{P}_{y}^{0.81}2{P}_{z}^{0.87}$ |

(*16e) | ${B}^{E{p}_{1}}={B}^{E{p}_{3}}-{11.0485}^{\text{}h}$ | ${|\mathsf{\psi}\rangle}_{BD(2)}=0.96S{P}_{{N}_{1}}^{1.0}+0.26S{P}_{B}^{{1.0}^{h}}$ | * 0.119 | $|\mathsf{\alpha}\rangle B:2{s}^{0.72}2{P}_{x}^{0.92}2{P}_{y}^{0.92}2{P}_{z}^{0.44}$ | |

${|\mathsf{\psi}\rangle}_{BD(3)}=0.96S{P}_{{N}_{1}}^{1.0}+0.26S{P}_{B}^{{1.0}^{h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}-{|\mathsf{\psi}\rangle}_{BD\left(2\right)}=0.77$ | $|\mathsf{\beta}\rangle N:2{s}^{0.77}2{P}_{x}^{0.87}2{P}_{y}^{0.87}2{P}_{z}^{0.86}$ | |||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}=0.42S{P}_{{N}_{1}}^{1.0}-0.90\text{}S{P}_{{B}_{2}}^{{1.0}^{h}}$ | * 0.028 | $|\mathsf{\beta}\rangle B:2{s}^{0.10}2{P}_{x}^{0.62}2{P}_{y}^{0.62}2{P}_{z}^{0.08}$ | |||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(2\right)}=0.26S{P}_{{N}_{1}}^{1.01}-0.96\text{}S{P}_{{B}_{3}}^{{0.95}^{h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}-{|\mathsf{\psi}\rangle}_{BD\left(2\right)}=0.7$ | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}=0.42S{P}_{{N}_{1}}^{1.0}-0.90\text{}S{P}_{{B}_{3}}^{{0.72}^{h}}$ | * 0.029 | ||||

${\tilde{\text{}A}}^{3}{\Pi}_{u}$ | ${N}^{E{p}_{2}}=-{18.5665}^{\text{}h}$ | $(5.19)$, $\text{}(5.19)$ | ${|\mathsf{\psi}\rangle}_{BD\left(1\right)}=0.91S{P}_{{N}_{1}}^{1.0}+0.40\text{}S{P}_{{B}_{2}}^{{1.62}^{h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=1.94$ | $|\mathsf{\alpha}\rangle N:2{s}^{0.75}2{P}_{x}^{0.81}2{P}_{y}^{0.60}2{P}_{z}^{0.86}$ |

(*18e) | ${B}^{E{p}_{1}}={B}^{E{p}_{3}}-{11.5955}^{\text{}h}$ | ${|\mathsf{\psi}\rangle}_{BD(2)}=0.97S{P}_{{N}_{1}}^{1.0}+0.26S{P}_{{B}_{2}}^{{1.0}^{h}}$ | * 0.107 | $|\mathsf{\alpha}\rangle B:2{s}^{0.74}2{P}_{x}^{0.09}2{P}_{y}^{0.70}2{P}_{z}^{0.43}$ | |

${|\mathsf{\psi}\rangle}_{BD(3)}=0.91S{P}_{{N}_{1}}^{1.0}+0.40S{P}_{{B}_{3}}^{{1.62}^{h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(4\right)}-{|\mathsf{\psi}\rangle}_{BD\left(1\right)}=1.01$ | $|\mathsf{\beta}\rangle N:2{s}^{0.80}2{P}_{x}^{0.87}2{P}_{y}^{0.90}2{P}_{z}^{0.86}$ | |||

${|\mathsf{\psi}\rangle}_{BD(4)}=0.70S{P}_{{B}_{2}}^{0.51}+0.70S{P}_{{B}_{3}}^{0.51}$ | * 0.064 | $|\mathsf{\beta}\rangle B:2{s}^{0.46}2{P}_{x}^{0.06}2{P}_{y}^{0.04}2{P}_{z}^{0.16}$ | |||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}=0.4S{P}_{{N}_{1}}^{1.0}-0.91\text{}S{P}_{{B}_{2}}^{{1.62}^{h}}$ | ${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(1\right)}-{|\mathsf{\psi}\rangle}_{BD\left(2\right)}=0.73$ | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(2\right)}=0.26S{P}_{{N}_{1}}^{1.0}-0.97\text{}S{P}_{{B}_{3}}^{{1.0}^{h}}$ | * 0.021 | ||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(3\right)}=0.40S{P}_{{N}_{1}}^{1.0}-0.91\text{}S{P}_{{B}_{3}}^{{1.62}^{h}}$ | |||||

${|\mathsf{\psi}\rangle}_{B{D}^{*}\left(4\right)}=0.70S{P}_{{B}_{2}}^{0.51}-0.70\text{}S{P}_{{B}_{3}}^{{051}^{h}}$ |

^{(a)}QCISD/EPR-III,

^{(h)}QCISD(T)/EPR-III.

**Table 3.**Isotropic Fermi contact coupling (IFCC) of B

_{2}${N}^{\left(0\right)}$ in ground (${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$) and exited (${\tilde{\text{}B}}^{4}{\prod}_{g}$) states.

State | $\Delta ={r}_{2}-{r}_{1}$ | $IFCC[f(\Delta )]$ | $\Delta \mathsf{\theta}={\mathsf{\theta}}_{1}-{\mathsf{\theta}}_{1}$ | $IFCC[f\left(\mathsf{\theta}\right)]$ |
---|---|---|---|---|

(*N_{e}) | ${\mathsf{\theta}}_{cons}=180.0$ | $N,{B}_{1}$,$\text{}{B}_{2}$ | ${r}_{cons}=1.3176$ | $N,{B}_{1}$,$\text{}{B}_{2}$ |

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ (*17e) | $\Delta \text{}=0.000$ | −29.81, 428.6, 428.6 | $\Delta \mathsf{\theta}=0.0$ | −29.81, 428.6, 428.6 |

$\Delta \text{}=0.010$ | −29.76, 386.2, 469.7 | $\Delta \mathsf{\theta}=2.0$ | −29.81, 428.71, 428.71 | |

$\Delta \text{}=0.020$ | −29.11, 348.2, 508.1 | $\Delta \mathsf{\theta}=3.0$ | −29.81, 428.77, 428.77 | |

$\Delta \text{}=0.030$ | −28.02, 316.1, 541.9 | $\Delta \mathsf{\theta}=5.0$ | −29.81, 428.95, 428.95 | |

$\Delta \text{}=0.040$ | −26.64, 290.3, 570.2 | $\Delta \mathsf{\theta}=10.0$ | −29.83, 429.84, 429.84 | |

$\Delta \text{}=0.050$ | −25.12, 270.7, 592.6 | $\Delta \mathsf{\theta}=20.0$ | −29.87, 433.44, 433.44 | |

$\Delta \text{}=0.060$ | −23.54, 256.7, 609.5 | $\Delta \mathsf{\theta}=30.0$ | −29.86, 439.68, 439.68 | |

$\Delta \text{}=0.070$ | −21.98, 247.6, 621.3 | $\Delta \mathsf{\theta}=40.0$ | −29.61, 448.75, 448.75 | |

$\Delta \text{}=0.080$ | −20.46, 242.7, 628.7 | $\Delta \mathsf{\theta}=50.0$ | −28.85, 460.82, 460.82 | |

$\Delta \text{}=0.082$ | −20.10, 242.1, 629.8 | $\Delta \mathsf{\theta}=60.0$ | −27.16, 475.86, 475.86 | |

$\Delta \text{}=0.085$ | −19.72, 241.6, 630.9 | $\Delta \mathsf{\theta}=70.0$ | −24.06, 493.31, 493.31 | |

$\Delta \text{}=0.086$ | 4.35, 60.3, 899.9 | $\Delta \mathsf{\theta}=80.0$ | −18.57, 512.55, 512.55 | |

$\Delta \text{}=0.087$ | 4.46, 59.4, 901.2 | $\Delta \mathsf{\theta}=90.0$ | −10.65, 533.75, 533.75 | |

$\Delta \text{}=0.090$ | 4.79, 56.6, 904.7 | |||

${\tilde{\text{}B}}^{4}{\prod}_{g}$ | $\Delta \text{}=0.000$ | 20.26, 377.8, 377.8 | ||

$\Delta \text{}=0.010$ | 20.26, 381.2, 377.9 | |||

$\Delta \text{}=0.020$ | 20.26, 384.5, 377.9 | |||

$\Delta \text{}=0.030$ | 20.28, 387.8, 377.9 | |||

$\Delta \text{}=0.040$ | 20.30, 391.2, 377.9 | |||

$\Delta \text{}=0.050$ | 20.30, 394.4, 377.9 | |||

$\Delta \text{}=0.060$ | 20.38, 397.8, 377.9 | |||

$\Delta \text{}=0.070$ | 20.44, 401.0, 377.8 | |||

$\Delta \text{}=0.080$ | 20.50, 404.27, 377.7 |

^{−1}, our calculations show that the total energies for both of them are the same (i.e., $-104.0781959$). This is due to the fact that the spin orbital energies are related to the small bending angles of A

_{1}and A

_{2}which have an extremely low bending frequency (70 cm

^{−1}). Harmonic frequencies were determined at the QCISD/EPR-III//prop=EPR and characterized by 228.79 cm

^{−1}(${\mathsf{\vartheta}}_{1}={\mathsf{\vartheta}}_{1}^{\prime},\text{}$ bending mode $\u201c{\mathsf{\pi}}_{u}\u201d$), 1178.64 cm

^{−1}(${\mathsf{\vartheta}}_{2}$, symmetric stretching “${\mathsf{\sigma}}_{g}$”) and 2146.42 (${\mathsf{\vartheta}}_{3}$, asymmetric stretching “${\mathsf{\sigma}}_{u}$”). The IR and Raman intensities for ${\mathsf{\vartheta}}_{3}$ are 10165.0 and 0.00, respectively, while the $\u201c{\mathsf{\vartheta}}_{2}\u201d$ mode has intensity in Raman (51.0) but zero intensity in the IR region.

^{−1}and 5834.79 cm

^{−1}, respectively. Those values are close to the photoelectron spectroscopy calculation results which Asmis et al. have shown [2]. They have discussed that the signal observed in the 355 nm and 266 nm photoelectron spectra of $\text{}{B}_{2\text{}}{N}^{-}$ has been indicated as due to a photodetachment from the ${\tilde{X}}^{1}\text{}{\Sigma}_{g}^{+}\text{})$ to the ground and lowest excited state of neutral B

_{2}N {${\tilde{X}}^{2}\text{}{\Sigma}_{u}^{+}\text{})$ and $(\text{}{\tilde{A}}^{2}\text{}{\Sigma}_{g}^{+}\text{})$ with a linear symmetry and assigned to the ${\tilde{X}}^{1}\text{}{\Sigma}_{g}^{+}\text{}\to {\tilde{X}}^{2}\text{}{\Sigma}_{u}^{+}+\text{}{e}^{-}$ and ${\tilde{X}}^{1}\text{}{\Sigma}_{g}^{+}\text{}\to {\tilde{A}}^{\text{}2}\text{}{\Sigma}_{g}^{+}+\text{}{e}^{-}$ transitions {the $(\text{}{\tilde{A}}^{2}\text{}{\Sigma}_{g}^{+}\text{})$ term energy T

_{0}is 0.785 eV or 6331.77 cm

^{−1}}.

^{−1}. The difference between (k − a) and (k − ${k}^{\prime}$) is $\approx $ 5 cm

^{−1}which is near 8.77 cm

^{−1}(different between ${C}_{2V}/{C}_{\infty V}$ and ${C}_{\infty V}/{D}_{\infty h}$ of strata/stratum). In addition ${\tilde{\text{}B}}^{2}{\prod}_{g}$ (exited state above the ${\tilde{A\text{}}}^{2}{\Sigma}_{g}^{+}$) is subject to the Renner-Teller effect, leading to a complicated pattern of bending vibrational levels. Our calculation shows the existence of a larger gap between $3{\mathsf{\sigma}}_{u}$ and $1{\pi}_{g}$ orbitals, thereby placing the transition to $\left[{A}^{\prime}\right]{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},1{\mathsf{\pi}}_{g}^{1}$ with ${\tilde{\text{}B}}^{2}{\prod}_{g}$ state much higher in energy. Analysis of the vibronic structure of the ${\tilde{\text{}B}}^{2}{\prod}_{g}-{\tilde{(X\text{}}}^{2}{\Sigma}_{u}^{+})$ band system shows the transition to ${\tilde{B}}^{2}{\prod}_{g}$ at 19,452 cm

^{−1}. Nevertheless, the nonbonding character of $1{\mathsf{\pi}}_{g}$ and $3{\mathsf{\sigma}}_{u}$ orbitals implies no significant change in B–N bond lengths in this transition, as it is observed by Ding et al. [56]. Therefore, in spite of ${\tilde{A}}^{2}{\Sigma}_{\left(g\right)}^{+}$, the “${\tilde{\text{}B}}^{2}{\prod}_{g}$” excited configuration does contribute to the ground state wave function as a subject of Renner-Teller effect.

_{3}. For even values of v

_{3}, this would be possible in principle by either dispersed fluorescence or stimulated emission pumping spectroscopy. In practice, however, it appears that these levels will be difficult to reach owing to poor Franck-Condon factors. For odd values of ${\mathsf{\vartheta}}_{3}$, a direct infrared absorption study provides the best method for the accurate measurement of those levels [56].

**v**, indicates a breakdown of the Franck–Condon (FC) approximation, it cannot be the only results from Herzberg–Teller vibronic coupling between the (${\tilde{X}}^{2}\text{}{\Sigma}_{u}^{+})$) and ($({\tilde{A}}^{2}\text{}{\Sigma}_{g}^{+})$) states involving the

_{3}**v**mode.

_{3}_{N}orbital is considered to be primarily core-like, forming the $3{\mathsf{\sigma}}_{g}$ orbital, though, of course, some mixing of the 2s

_{N}orbital into the other ${\mathsf{\sigma}}_{g}$ orbitals ix expected. NBO analysis of the orbital containing the unpaired electron in BNB shows that most of the spin density is located in the boron sp orbitals. The boron atomic orbitals are best described as a “sp” hybrid, directed away from the nitrogen atom however, bonding with respect the $\mathsf{\sigma}P$ orbital on the nitrogen atom. The $2{\mathsf{\sigma}}_{u}$ orbital is a bonding combination of the $2p\mathsf{\sigma}$ orbital on the central nitrogen atom with $2sp\mathsf{\sigma}$ hybrid orbitals on the two borons. The $1{\mathsf{\pi}}_{u}$ orbital is a bonding combination of all $2p\mathsf{\pi}$ orbitals on all three atoms, while $\u201c4{\mathsf{\sigma}}_{g}\u201d$ and $\u201c3{\mathsf{\sigma}}_{u}\u201d$ orbitals are close lying and not strongly bonding in character.

**g**” tensor for BNB shows the large boron isotropic amount. Thus, the properties of the

**g**tensor eliminate the possibility of a low exited 2Πg state for the radical. The averaged vibrations of B–N bond lengths, $IFCC\left[f(\Delta )\right]$ and $IFCC\left[f\left(\mathsf{\theta}\right)\right]$ can be shown, even if the ground state is precisely linear, quasi-linear or the geometry at the potential minimum went through a symmetry breaking to form a ${C}_{\infty v}$ structure.

^{−1}[9,22,23] or 20 cm

^{−1}[20], but also it creates several SB through the asymmetry stretching (or interaction between asymmetry stretching and bending) with different barrier energies from high to small values (about 5 cm

^{−1}) (Figure 1).

**Figure 2.**Relative energies of B

_{2}N

^{(−, 0, +)}vs. B-N-B bond angle in various level of methods for anion, cation and radical forms of BNB respectively

_{2}N

^{(−, 0)}in our calculations has not been observed, though, for cation, there is a bulge in the curve at 90° in MP

_{4}DQ and MP

_{4}DSQ methods which indicates a cyclic B

_{2}N

^{+}.

**Figure 3.**Isotropic Fermi contact coupling (IFCC) of B

_{2}N

^{(0)}: (

**a**) Function of angles changing; (

**b**) Function of distances changing. The axes for nitrogen and borons have different scales; (

**a'**) IFCC for Nin a short scale between −30 to −29.8 indicates of IFCC breaking and (

**a''**) different IFCC of B and N in a short scale.

_{4}DQ, MP

_{4}DSQ and HF/aug-cc-pVTZ calculations (Figure 2) the linear structure clearly has the lowest energies for radical and anion structures. Martin [7] has shown a cyclic B

_{2}N (

^{2}B

_{2}) via reactions of pulsed laser produced boron and nitrogen atoms in a condensed argon stream (at higher laser power reactions) and has discussed that the vibration 882.3 cm

^{−1}must be considerably an-harmonic. This possibility receives substantial support from the five combination bands observed in the 3000–6500 cm

^{−1}regions. The 882.3 cm

^{−1}one is assigned to the anti-symmetric B-N stretching fundamental v

_{3}(b

_{2}) of cyclic B

_{2}N and the 1998.4 cm

^{−1}combination band is the sum of v

_{1}(a

_{1}), the symmetric B-N stretching fundamental, and v

_{3}. The difference 1998 − 882 = 1116 cm

^{−1}can help to measure the v

_{1}.

_{g}→ σ

_{u}excitation is less than 6000 cm

^{−1}, which indicates that the higher overtones of the cyclic B

_{2}N (

^{2}B

^{2}state) vibrations will display significant vibronic interaction effects. The failure to observe cyclic B

_{2}N in the

^{2}B

^{2}state by ESR [4] is most likely due to the differences in production and relaxation of the energized evaporated species. The fact that the radical BNB might be converted to cation BNB towards the cyclic B2N (due to the laser ionization effect) can be predicted from Martin’s study. In addition, Becker et al. [58] used laser ionization mass spectrometry to study the formation of ${B}_{n}{N}_{m}^{+}$ clusters ions in laser plasma which resulted in production of BNB cation whereas our calculations resulted in production of cyclic B

_{2}N

^{+}.

_{2}${N}^{\left(0\right)}$ is stabilized by six resonance structures of the linear forms and three resonance structures of cyclic forms (with various distributions) as follows:

_{2}${N}^{\left(0\right)}$ with different levels of theory, nitrogen is always positive and close to zero while the two boron atoms are always negative near zero and the converse appears for the B

_{2}N

^{(+)}species. The atomic charges via 4516 points for one of the ESP fitting at QCISD/EPR-III level of theory for radical shows the values of 0.076055, −0.037580 and −0.038476 for N and two boron atoms, respectively.

^{2}Σ

^{+}: $\text{}1{\mathsf{\sigma}}^{2},2{\mathsf{\sigma}}^{2},3{\mathsf{\sigma}}^{2},4{\mathsf{\sigma}}^{2},5{\mathsf{\sigma}}^{2},6{\mathsf{\sigma}}^{2},7{\mathsf{\sigma}}^{1},1{\mathsf{\pi}}^{2},1{\mathsf{\pi}}^{2}$ and

^{6}Σ

^{+}: $:1{\mathsf{\sigma}}^{2},2{\mathsf{\sigma}}^{2},3{\mathsf{\sigma}}^{2},4{\mathsf{\sigma}}^{2},5{\mathsf{\sigma}}^{2},6{\mathsf{\sigma}}^{2},1{\mathsf{\pi}}^{2},7{\mathsf{\sigma}}^{1},1{\mathsf{\pi}}^{2}$ the dynamic correlation would be even smaller [1]. With a cc-pVQZ basis set and high correlation, the single reference CCSD (T) energy gap between the symmetric and asymmetric configurations is 136 cm

^{−1}[23] which is reduced to 99 cm

^{−1}in the RMR CCSD (T) method [14]. An additional extended discussion would be reported in a subsequent publication concerning strata/stratum (S/s) configuration.

_{2}${N}^{(-)}$ should be a

^{3}B

_{2}state and not a single state.

^{−1}(${\mathsf{\vartheta}}_{1}={\mathsf{\vartheta}}_{1}^{\prime},\text{}$ bending mode “${\mathsf{\pi}}_{u}$”) and 1203.42 cm

^{−1}(${\mathsf{\vartheta}}_{2}$, symmetric stretching “${\mathsf{\sigma}}_{g}$”) and 1837.50 (${\mathsf{\vartheta}}_{3}$, asymmetric stretching “${\mathsf{\sigma}}_{u}$”). The IR and Raman intensities for ${\mathsf{\vartheta}}_{3}$ are 1494.6 and 0.00, respectively, while the ${\mathsf{\vartheta}}_{2}$” mode has intensity in Raman (39.97), but zero intensity in IR region. The valence orbital occupancy of ground state (${\tilde{X\text{}}}^{1}\text{}{\Sigma}_{g}^{+})$ is: $1{\mathsf{\sigma}}_{g}^{2},1{\mathsf{\sigma}}_{u}^{2},2{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{g}^{2},2{\mathsf{\sigma}}_{u}^{2},1{\mathsf{\pi}}_{u}^{4},4{\mathsf{\sigma}}_{g}^{2},3{\mathsf{\sigma}}_{u}^{2}$, while the lowest excited triplet state in the ${D}_{\infty h}$ representation for anion form has calculated (Table 1) and is predicted to be a ${\tilde{\text{}A}}^{3}{\Pi}_{u}$ state, lying 2.94 eV {${\tilde{\text{}A}}^{3}{\Pi}_{u}\left(a\right)-{\tilde{X}}^{1}{\Sigma}_{g}^{+}\left(a\right)$ in Table 1} above the ${\tilde{X\text{}}}^{1}{\Sigma}_{g}^{+}$ state (with the calculated value of 2.7 eV anion with photoelectron spectroscopy of B

_{2}N

^{(−)}).

_{2}${N}^{(+)}$ in which the nitrogen in cation is negative and two borons are positive in all ESP calculations. Both of the two highest occupied orbitals are predominantly nonbonding with the electron density localized mainly on the “terminal” boron atoms.

_{2}N

^{(0)}as a function of angles and distances is shown in Table 4 and Figure 3. Although the symmetry breaking cannot be seen in the IFCC$\left\{f\left(\Delta r\right)\right\}$, it can be seen in the IFCC$\left\{f\left(\Delta \mathsf{\theta}\right)\right\}$ for the nitrogen in range of angles between $\Delta \mathsf{\theta}=$ (50, −50) (Figure 3a'). On the other hand, the SB has not been observed for the $\Delta \left(\Delta \mathsf{\theta}\right)$ = $\Delta {\mathsf{\theta}}_{B}-\Delta {\mathsf{\theta}}_{N}$ in range of $\Delta \mathsf{\theta}=$ (5, −5) (Figure 3a''). It seems that nitrogen in SB problem plays a major role and it depends on asymmetry bond changing and angle deformation interaction $({\mathsf{\pi}}_{u}+{\mathsf{\sigma}}_{u})$.

**Table 4.**

**{**TQ

^{#}: Traceless Quadrupole moment (Debye-Ang)}; (

**§**) Charges from ESP fitting and Isotropic Fermi Contact Couplings (MHz) (IFCC).

State | ${E}_{e}\left(Hartree\right)$ | ${B}^{\mathsf{\delta}{q}_{1}}\u2013{N}^{\mathsf{\delta}{q}_{2}}\u2013{B}^{\mathsf{\delta}{q}_{3}^{(\S )}}$ | $T{Q}_{xx}^{\#}$ |
---|---|---|---|

(* N_{e}) | IFCC(N,B_{1},B_{2}) | $T{Q}_{yy}$ | |

$T{Q}_{zz}$ | |||

$\tilde{X\text{}}$ ^{2}Σ^{+} | $-104.0781959{\text{}}^{a}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.015{\text{}}^{a}$ | $-1.8265{\text{}}^{a}$ |

$({C}_{\infty v})$ | $-104.0820328{\text{}}^{{a}^{\prime}}$ | $\mathsf{\delta}{q}_{2}=0.03{\text{}}^{a}$ | $0.8644{\text{}}^{a}$ |

(*17e) | $-104.0754917{\text{}}^{{a}^{\text{'}\text{'}}}$ | $-29.8,\text{}427.2,\text{}{429.6}^{\text{}a}$ | $0.9621{\text{}}^{a}$ |

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-104.0781959\text{}{\text{}}^{b}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.015{\text{}}^{b}$ | $0.9621{\text{}}^{b}$ |

$({D}_{\infty h})$ | $\mathsf{\delta}{q}_{2}=0.03{\text{}}^{b}$ | $0.9621{\text{}}^{b}$ | |

$-104.159145{\text{}}^{f}$ | $-29.9,\text{}428.2,\text{}{428.2}^{\text{}b}$ | $-{1.9241}^{\text{}b}$ | |

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-{103.6396773}^{\text{}d}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.0165{\text{}}^{d}$ | ${Q}_{xx}=-{16.5}^{\text{}d}$ |

$({D}_{\infty h})$ | $\mathsf{\delta}{q}_{2}={0.033}^{d}$ | ${Q}_{yy}=-16.5{\text{}}^{d}$ | |

(*17e) | ${Q}_{zz}=-19.8{\text{}}^{d}$ | ||

${\tilde{X\text{}}}^{2}{\Sigma}_{u}^{+}$ | $-104.159145{\text{}}^{f}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.0045{\text{}}^{f}$ | $0.8928{\text{}}^{f}$ |

$({D}_{\infty h})$ | $-{104.1355512}^{\text{}n}$ | $\mathsf{\delta}{q}_{2}=0.009$ | $0.8928$ |

$-1.7855$ | |||

${\tilde{A\text{}}}^{2}$ | $-104.1047582{\text{}}^{k}$ | $137.9,\text{}817.0,\text{}817.0{\text{}}^{u}$ | $-17.87{\text{}}^{u}$ |

(*17e) | $-104.0781729{\text{}}^{{K}^{\prime}}$ | $-17.87{\text{}}^{u}$ | |

$-20.80{\text{}}^{u}$ | |||

${\tilde{\text{}B}}^{4}{\prod}_{g}$ | $-104.0297022{\text{}}^{h}$ | $20.26,\text{}377.84,\text{}377.84{\text{}}^{h}$ | 1.0334 |

(*17e) | $-104.0141196{\text{}}^{a}$ | -2.4251 | |

1.3917 | |||

${\tilde{X\text{}}}^{1}$ | $-104.1965676{\text{}}^{a}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.934$ | $-14.8029{\text{}}^{a}$ |

(*18e) | $-104.2019114{\text{}}^{{a}^{\prime}}$ | $\mathsf{\delta}{q}_{2}=0.868{\text{}}^{a}$ | $7.4015{\text{}}^{a}$ |

$-104.1950169{\text{}}^{{a}^{\text{'}\text{'}}}$ | −14.98, 211.91, 211.91 | $7.4015{\text{}}^{a}$ | |

${\tilde{X\text{}}}^{1}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.934$ | $7.4015{\text{}}^{b}$ | |

(*18e) | $-104.1965676{\text{}}^{b}$ | $\mathsf{\delta}{q}_{2}={0.868}^{\text{}b}$ | $7.4015{\text{}}^{b}$ |

$-14.803{\text{}}^{b}$ | |||

${\tilde{X\text{}}}^{1}$ | $-104.1145486{\text{}}^{c}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.987$ | $-15.7996{\text{}}^{c}$ |

(*18e) | $-{104.1162881}^{{c}^{\prime}}$ | $\mathsf{\delta}{q}_{2}=0.974{\text{}}^{c}$ | $7.8992{\text{}}^{c}$ |

$-104.112568{\text{}}^{{c}^{\text{'}\text{'}}}$ | $7.9004{\text{}}^{c}$ | ||

${\tilde{\text{}A}}^{3}{\Pi}_{u}$ | $-104.0884685{\text{}}^{a}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=-0.65{\text{}}^{h}$ | $2.444{\text{}}^{h}$ |

(*18e) | $-104.1164696{\text{}}^{h}$ | $\mathsf{\delta}{q}_{2}=0.3$ | $6.5747$ |

$-104.0809447{\text{}}^{{a}^{\prime}}$ | $-9.0187$ | ||

${\tilde{X\text{}}}^{1}$ | $-103.7454365{\text{}}^{a}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=0.805{\text{}}^{a}$ | ${8.0329}^{\text{}a}$ |

(*16e) | $-103.8054934{\text{}}^{{a}^{\prime}}$ | $\mathsf{\delta}{q}_{2}=-0.611$ | $-4.0164$ |

$-{103.7545006}^{\text{}{a}^{\prime \prime}}$ | $-4.0164$ | ||

${\tilde{X\text{}}}^{1}$ | $-103.6023122{\text{}}^{m}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=0.828{\text{}}^{f}$ | $7.009{\text{}}^{m}$ |

(*16e) | $-103.3012055{\text{}}^{g}$ | $\mathsf{\delta}{q}_{2}=-0.656{\text{}}^{f}$ | $-3.505{\text{}}^{m}$ |

$-103.8377401{\text{}}^{f}$ | $-3.505{\text{}}^{m}$ | ||

$\tilde{B\text{}}$ ^{3}Σ_{g} | $-103.7609202{\text{}}^{a}$ | $\mathsf{\delta}{q}_{1}=\mathsf{\delta}{q}_{3}=0.763{\text{}}^{a}$ | $-3.6992{\text{}}^{a}$ |

(*16e) | $-103.7767141{\text{}}^{h}$ | $\mathsf{\delta}{q}_{2}=-0.525{\text{}}^{a}\text{}$ | $-3.6992$ |

$-103.7628505{\text{}}^{{a}^{\prime}}$ | $7.3983$ |

^{(a)}QCISD/EPR-III;

^{(d)}CASSCF(11,12)/UHF;

^{(a}

^{')}MP

_{4}D/EPR-III//QCISD/EPR-III;

^{(m)}CASSCF(10,12)/EPR-II;

^{(a}

^{'')}MP

_{4}SDQ/EPR-III//QCISD/EPR-III;

^{(g)}CASSCF(10,12)rohfAUG-cc-pvqz;

^{(b)}QCISD/EPR-III;

^{(c)}QCISD/EPR-II;

^{(c}

^{')}MP

_{4}D/EPR-II//QCISD/EPR-II;

^{(c}

^{'')}MP

_{4}SDQ/EPR-II//QCISD/EPR-II;

^{(f)}CBS-lq;

^{(h)}QCISD(T)/EPR-III;

^{(n)}CBS4O;

^{(u)}TD/EPR-II;

^{(k)}TD/EPR-III//QCISD(T)/EPR-III;

^{(K}'

^{)}TD/EPR-III//QCISD/EPR-III.

^{3}Σ

_{g}for cation). In other words, the quadrupole moment for BNB is highly sensitive to angle deformation and bond distance changing, so it can be discussed for any SBs in terms of quadrupole moment.

**Figure 4.**Parameters of Charge correction coefficients, vs. changing of boron and nitrogen distances for B

_{2}N

^{(−, 0, +)}for anion , cation and radical.

## 5. Conclusions

_{2}N

^{(−, 0, +)}in Born–Oppenheimer approximation mostly depends on the variables such as B-N bond length, using large and larger basis sets and more and more electron correlation are doomed to result in wrong limit for the energy level of SB barriers or SB estimation. It is prudent to employ a highly correlated method which can use a large number of reference determinants to recover dynamic and static correlations. We have shown that the SB is generally applied to the failure of the electronic wave function in order to be transformed as an irreducible representation of the molecular point group, so the failure of the electronic wave function is purely artifactual. It is not wise to conclude that the only special level of theory on the symmetry breaking for BNB is real (which has been concluded in reference [9]) because there exist some hidden variables in the electronic wave functions which should be considered. We have found that the symmetry breaking (SB) for some hidden variables (such as charge distribution) not only exhibit an energy barrier, it also creates several SBs through the asymmetry stretching with bending mode interaction.

## Author Contributions

## Conflicts of Interest

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Monajjemi, M.; Bagheri, S.; Moosavi, M.S.; Moradiyeh, N.; Zakeri, M.; Attarikhasraghi, N.; Saghayimarouf, N.; Niyatzadeh, G.; Shekarkhand, M.; Khalilimofrad, M.S.; Ahmadin, H.; Ahadi, M. Symmetry Breaking of B_{2}N^{(−, 0, +)}: An Aspect of the Electric Potential and Atomic Charges. *Molecules* **2015**, *20*, 21636-21657.
https://doi.org/10.3390/molecules201219769

**AMA Style**

Monajjemi M, Bagheri S, Moosavi MS, Moradiyeh N, Zakeri M, Attarikhasraghi N, Saghayimarouf N, Niyatzadeh G, Shekarkhand M, Khalilimofrad MS, Ahmadin H, Ahadi M. Symmetry Breaking of B_{2}N^{(−, 0, +)}: An Aspect of the Electric Potential and Atomic Charges. *Molecules*. 2015; 20(12):21636-21657.
https://doi.org/10.3390/molecules201219769

**Chicago/Turabian Style**

Monajjemi, Majid, Samira Bagheri, Matin S. Moosavi, Nahid Moradiyeh, Mina Zakeri, Naime Attarikhasraghi, Nastaran Saghayimarouf, Ghorban Niyatzadeh, Marzie Shekarkhand, Mohammad S. Khalilimofrad, Hashem Ahmadin, and Maryam Ahadi. 2015. "Symmetry Breaking of B_{2}N^{(−, 0, +)}: An Aspect of the Electric Potential and Atomic Charges" *Molecules* 20, no. 12: 21636-21657.
https://doi.org/10.3390/molecules201219769