^{1}

^{2}

^{1}

^{*}

The potential energy curves have been investigated for the 40 lowest electronic states in the
^{2s+1}Λ
^{(±)}representation below 25000 cm
^{-1} of the molecule NiO via CASSCF, MRCI (single and double excitation with Davidson correction) and CASPT2 methods. The harmonic frequency
ω_{e} , the internuclear distance
r_{e}, the rotational constant
B_{e}, the electronic energy with respect to the ground state
T_{e}, and the permanent dipole moment
μ have been calculated. By using the canonical functions approach, the eigenvalues
E_{v}, the rotational constant
B_{v} and the abscissas of the turning points
r_{min} and
r_{max} have been calculated for the considered electronic states up to the vibration level
v = 12. Eleven electronic states have been studied theoretically here for the first time. The comparison of these values to the theoretical and experimental results available in literature shows a very good agreement.

The metal oxide NiO shows complicated electronic spectra because of the presence of large number of electronic states derived from several low-lying configurations but it gives a systematic example of chemical bonding, which depends on the relative energy between the 3d orbital of the metal and the 2p orbital of oxygen [1,2]. The transition metal oxides have interesting applications in many fields such as materials application and the oxidation of metal surfaces. Among these compounds the NiO molecule which is considered as a prototype of ionic crystals, it is classified as a Mott-Hubbard insulator of very low conductivity. The conductivity of nanostructred NiO was found to be enhanced by six to eight orders of magnitude over those of NiO single crystals [3-10]. The magnetic properties of different sizes of NiO nanoparticles reveal the presence of superparamagnetism as evidenced by the increasing magnetization with decreasing size as well as the magnetic hysteresis at low temperatures. Nanomagnetism promises have applications in magnetic storage with nanomagnetic particles, improved battery lifetimes and also quantum computing [11-14]. The nanoarticles formed by the NiO molecule have many applications in electronics, optical, electro-optical devices and photocatalytic reaction. Despite this importance of the nickel oxide NiO, this molecule has been studied experimenttally and theoretically [15-35] where a limited number of electronic states have been obtained with the corresponding molecular constants. The theoretical calculation of the NiO molecule is an extreme computational challenge because of the degeneracy of several energetically low-lying excited states and the open d, p, and s shells. The presence of the d shell implies large multiplicities which are split by spin-orbit interaction. The components of the spin and the many states perturb each other. The prediction and assignment of the electronic configuration in the ground and excited states and the description of the bonding may often be difficult.

Based on our previous theoretical calculation [36-45], the important connection between energy relations of solids and molecules [^{–1}. In this work, we investigate the potential energy curves (PECs), the electric dipole moment and spectroscopic constants for the 40 ^{2s+1}Λ^{(±)} low-lying electronic states of this molecule obtained by MRCI and RSPT2 calculations. Taking advantage of the electronic structure of the investigated electronic states of the NiO molecule and by using the canonical functions approach [_{v}_{, }the rotational constant B_{v} and_{ }the abscissas of the turning points r_{min }and r_{max} have been calculated for several vibrational levels of the considered electronic states.

The PECs of the lowest-lying electronic states of the NiO molecule have been investigated via CASSCF and CASPT2 methods. The MRCI calculations (single and double excitations with Davidson corrections) were performed. The Nickel atom is treated as a system with 10 inner electrons taken into account using the basis LANL2DZECP [_{2}-Xfit [

In the representation ^{2s+1}Λ^{(±)}, 40 electronic states have been investigated for 46 internuclear distances in the range 1.331 Å ≤ r ≤ 2.681 Å by using the MRCI and RSPT2 calculations. The potential energy curves for the singlet and triplet states, obtained by MRCI calculation, are given in Figures 1-4.

The spectroscopic constants such as the vibration harmonic constant, 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. These values are given in

An overlap between the 3d and 4s orbitals of the Ni atom and the 2p orbitals of the oxygen atom lead to the formation of the molecular orbitals of the molecule NiO. The ground state of this molecule is confirmed theoreticcally and experimentally to be a^{3}S^{–} [1,16-23,34-35, 52-58], its electronic configuration can approximately described as 8σ^{2} 3π^{4} 1δ^{4} 9σ^{2} 4π^{2}. The 9σ and 4π orbitals are antibonding, the 1δ orbital is nonbonding, and the

remaining orbitals are bonding [^{2+}O^{2–} zero-order character at low energy, while the NiO molecule has the chance to be represented by Ni^{+}O^{–} atomic-ion-in-molecule model [_{e} and the vibrational harmonic constant, for the ground state, by using the MRCI calculation, are in very good agreement with the experimental values in literature with the relative differences 2.7% (Ref. [_{e}/r_{e} < 3.8% (Ref. [_{e}/r_{e} < 8.3% (Ref. [_{e} with those calculated theoretically in literature we obtained a very good agreement with relative differences 3.8% (Ref. [_{e}/r_{e} < 5.6% (Ref. [_{e}/r_{e} < 8.3% (Ref. [^{–1} and our values are in agreement with the lower value.

A large number of low-lying electronic states, many with very high spin multiplicity, of 3d metal oxides are produced by the unpaired electrons, therefore strong state mixing results in spectroscopy which implies difficult theoretical calculation. Another dimension of complexity can be added originating for the large nuclear spin and magnetic moment of the nuclei with odd atomic number. The first excited state of the NiO molecules has long been a matter of controversy. The theoretical calculation of Bauschlicher Jr. and collaborators [21,22,56] and that of Bakalbassis et al. [^{3}P. Moravec and jarrold [^{3}P is the first excited state. For Ram and Bernath [^{3}P state is located experimentally above the ground state X^{3}S^{–} at 4330 and 4293 cm^{–1} respectively. From the photoelectron spectroscopy technique, Ramond et al. located the ^{3}P_{2} and ^{3}P_{1} at 3916 and 4327 cm^{–1} respectively [^{1}P rather than ^{3}P and located this state at 6000 cm^{–1} above the ground state X^{3}S^{–}. This prediction of Walch and Goddard [^{1}Δ is in agreement with the present work but our ^{3}P state is located at 5068.6 cm^{–1} above the ground state. Moreover between the first excited state (1)^{1}Δ and the (1)^{3}P we found the 4 new singlet electronic states (1)^{1}Δ, (2)^{1}Δ, ^{1}Φ or ^{1}P and (3)^{1}Δ by using MRCI calculation and the 2 new triplet electronic states (1)^{3}Γ, (1)^{3}Φ, and the 3 new singlet electronic states ^{1}P (1)^{1}Γ and ^{1}Δ by using RSPT2 calculation. The agreement between the 2 ways of calculation is only for the last state ^{1}Δ. Ramond et al. [^{–1} between X^{3}S^{–} and ^{3}P_{2} states without assigning the nature of this state. By comparing the data of this state with our calculated values, one can find that, it is either (2)^{1}Δ or (1)^{3}Φ.

Also without assigning the name of the state, Ramond et al. [^{–1} above the ^{3}P_{1}, this energy is in very good agreement with the energy of our calculated (2)^{3}Φ state with the relative differences ΔT_{e}/T_{e} = 0.8% and Δr_{e}/r_{e} = 5%. Between the values of energy 7659.2 cm^{–1} < T_{e} < 8788.4 cm^{–1}, Ramond et al. [^{3}Φ_{i}, ^{3}Δ_{i} and ^{3}P_{i}. By comparing this data to our calculated values in the present work (^{3}Φ_{I} and ^{3}Δ_{I} are excluded since T_{e}((2)^{3}Φ = 5271.9 cm^{–}^{1}), T_{e}((3)^{3}Φ = 13911.9 cm^{–}^{1}), T_{e}((1)^{3}Δ = 10185.1 cm^{–}^{1}), and T_{e}((2)^{3}Δ = 10243.65 cm^{–}^{1}), therefore the only possible assignment is (2)^{3}P. These values of energy obtained by Ramond et al. [^{1}Δ in ^{1}P the energy T_{e} = 10095.1 cm^{–1} which fits with our calculated value of the state (2)^{1}P with the relative difference ΔT_{e}/T_{e} = 6.7%.

The values of the energy T_{e} of the states ^{3}Φ_{2,3,4} and ^{1}Δ obtained experimentally by Moravec and Jarrold [^{3}Φ and (4)^{1}Δ of the present work. Our calculated value of for the state (4)^{1}Δ is in good agreement with that measured by Moravec and Jarrold [^{3}P_{1,2,3} states detected by Moravec and Jarrold [^{3}P of the present work (

The assignment of the three states a(^{1}Δ), b(^{1}S^{+}) and c(^{1}∏) detected experimentally by Wu and Wang [^{1}Δ, ^{1}S^{+} and (3)^{1}P with the relative differences ΔT_{e}/T_{e}_{ }(a(^{1}Δ)) = 6%, ΔT_{e}/T_{e} (b(^{1}S^{+})) = 0.5%, ΔT_{e}/T_{e} (c(^{1}P)) = 1.3% respectively. Baushlicher Jr. [^{1}S^{+} state at 8340 cm^{–1} above the ^{3}S^{–} which is in disagreement with our calculated value and those of Wu and Wang [_{e}, in the present work, for the (2)^{3}Φ and (3)^{3}Φ states are respectively 5271.9 and 13911.9 cm^{–1}, therefore the possible assignment of the B(^{3}Φ) state, where T_{e} = 10025.7 cm^{–1}, investigated by Wu and Wang [^{3}Φ (_{e} and by bakalbassiss et al. [^{ 3}Δ and ^{1}S^{+}, which are fitting with our (4)^{3}Δ and ^{1}S^{+}, are in disagreement with the values calculated in the present work, while there are good agreements between our data and those of Bauschlicher Jr. and Maitre [_{e} for these states with the relative differences Δr_{e}/r_{e}(^{3}Δ) = 9% and Δr_{e}/r_{e}(^{1}S^{+}) = 5%.

The ^{3}S^{–} at 16,000 cm^{–1} predicted by Friedman-Hill and Field [^{3}S^{–} state calculated in the present work with the relative difference ΔT_{e}/T_{e} = 0.3%. The detected ^{3}P state by Walch and Goddard [^{–1} fits with our assignment of the state (2)^{3}P where the relative difference in energy ΔT_{e}/T_{e} = 12%. The comparison of our results with the most recent investigation on the molecule NiO [^{3}S^{–}, but our assumption for the energies 19452.6 cm^{–1} and 19447.3 cm^{–1} are in agreement with that given by Wu and Wang [

The electric dipole moment is an effective gauge of the ionic characters; it is helping for understanding the macroscopic properties of imperfect gases, liquids and solids and is of great utility in the construction of molecular orbital. The expectation value of this operator is sensitive primarily to the nature of the least energetic and most chemically relevant valence electrons. To understand the ionic behavior of the excited electronic states we have presented in Figures 5-8 the adiabatic permanent dipole moment for the investigated electronic states in the range of the considered internuclear distance. It was seen that the variation of the adiabatic dipole moment is very important in the vicinity of the avoided crossings and the position of the peaks corresponds to the positions of these avoided crossings of the adiabatic curves. Each time an adiabatic state loses its ionic character, it becomes neutral and the corresponding dipole moment tends towards zero. One can notice the fit between the positions of the intersections of the permanent dipole moment curves and the position of the avoided crossings of the corresponding potential energy curves for the following states (4)^{1}Δ/(5)^{1}Δ (6)^{1}Δ/(7)^{1}Δ and (1)^{3}Δ/(3)^{3}Δ at 1.631 Å, 1.812 Å and 1.811 Å respectively. These fittings confirm the validity and the accuracy of the investigated data in the present work on the molecule NiO.

Within the Born-Oppenheimer approximation, the vibration rotation motion of a diatomic molecule in a given electronic state is governed by the radial Schrödinger equation

where r is the internuclear distance, v and J are respecttively the vibrational and rotational quantum numbers, and are respectively the eigenvalue and the eigenfunction of this equation. In the perturbation theory these functions can be expanded as

with e_{0} = E_{v}, e_{1} = B_{v}, e_{2} = –D_{v}, f_{0} is the pure vibration wave function and f_{n} its rotational corrections. By replacing Equations (2) and (3) into Equation (1) and since this equation is satisfied for any value of l, one can write [47,59-63]

···

where, the first equation is the pure vibrational Schrödinger equation and the remaining equations are called the rotational Schrödinger equations. One may project Equations (7) onto f_{0} and find

···

Once e_{0} is calculated from Equation (4), e_{1}, e_{2}, e_{3}··· can be obtained by using alternatively Equations (5) and (6). By using the canonical functions approach [_{v}, the rotational constant B_{v}, the distortion constant D_{v}, and the abscissas of the turning point r_{min} and r_{max} have been calculated up to the vibrational levels v = 12 for the investigated electronic states in the present work. These values for the state X^{1}∑^{+} and the (2)^{3}Φ (as illustration) are given in _{v} and D_{v} in literature [18,19,25] with our calculated values for these states showed a good agreement with relative difference ΔB_{v}/B_{v} ≈ 8% for v = 0, 1 and ΔD_{v}/D_{v} equal 7.5% and 2.4 % respectively for v = 1 and v = 2.

In the present work, the ab initio investigation for the 40 low-lying singlet and triplet electronic states of the NiO molecule has been performed via CAS-SCF/MRCI and CASPT2 methods. The potential energy and the dipole moment curves have been determined along with the spectroscopic constants T_{e}, r_{e}, , and the rotational constant B_{e} for the lowest-lying electronic states. The comparison of our results, for different states, with those obtained experimentally and theoretically shows a good agreement. By using the canonical functions approach [47,59-63], the eigenvalue E_{v}, the rotational constant B_{v},

and the abscissas of the turning points r_{min} and r_{max} have been calculated up to the vibrational level v = 12. Eleven electronic states have been investigated in the present work for the first time.