MOLECULAR THEORY OF HYDROGEN MOLECULE BY LCAO-MO METHOD

September 20, 2022
by Bhoomika Sheladiya

INTRODUCTION

WHAT IS MOLECULAR ORBITAL THEORY?

MO = Molecular Orbital (Abbreviation)

In molecular orbital theory, the electrons in a molecule are viewed as moving under the influence of the atomic nuclei throughout the entire molecule rather than being allocated to specific chemical bonds between atoms.

Quantum mechanics characterizes the spatial and energetic characteristics of electrons in the molecular orbitals that encircle and contain valence electrons between atoms in molecules.

According to Pauli’s principle, each MO holds one or two electrons. With the help of MO theory can explain Paramagnetism of the molecules like O2 and NO. The molecular orbital theory is made by applying the DFT (density functional theory). Also, the importance of molecular quantum mechanics is the Born – Oppenheimer approximation.

WHAT IS LCAO (LINEAR COMBINATION OF ATOMIC ORBITALS)?

Hund and Mulliken give an approach to the electrons in atoms occupying atomic orbital, described as the region where the high possibility of finding the electrons is called LCAO. (LINEAR COMBINATION OF ATOMIC ORBITALS)

WHAT IS BMO (BONDING MOLECULAR ORBITALS)?

When forming two molecular orbitals, it is an overlapping of two atomic orbitals one has at a lower energy level than the overlapping of atomic orbital is known as BMO.

WHAT IS ABMO (ANTIBONDING MOLECULAR ORBITALS)?

Two molecular orbitals it is overlapping of two atomic orbitals; one lies at a high energy level called ABMO.

MOLECULAR THEORY OF H2 MOLECULE BY LCAO-MO METHOD

Homonuclear diatomic, H2 molecule, the MO ψ written as,

\[\displaystyle \psi ={{c}_{1}}{{\phi }_{1}}+{{c}_{2}}{{\phi }_{2}}----(1)\]

Here, Φ1 and Φ2  = Atomic orbitals centered on hydrogen atoms, 1 &2

C1 and c2 = atomic orbitals coefficient (such as S, P, d….)

By using the Schrodinger wave equation, the E of the MO can be,

\[\displaystyle \hat{H}\psi =E\psi ----(2)\]

The MOs are unnormalized; the energy E is,

\[\displaystyle E=\frac{{\int{{{{\psi }^{*}}\hat{H}\psi d\tau }}}}{{\int{{{{\psi }^{*}}\psi d\tau }}}}----(3)\]

If ψ* is real, then ψ*= ψ

Now, substituting equation (1) value into the equation (3), we get,

\[\displaystyle E=\frac{{\int{{\left( {{{c}_{1}}{{\phi }_{1}}+{{c}_{2}}{{\phi }_{2}}} \right)\hat{H}\left( {{{c}_{1}}{{\phi }_{1}}+{{c}_{2}}{{\phi }_{2}}} \right)d\tau }}}}{{\int{{{{{\left( {{{c}_{1}}{{\phi }_{1}}+{{c}_{2}}{{\phi }_{2}}} \right)}}^{2}}d\tau }}}}\]
\[\displaystyle =\frac{{\int{{\left( {{{c}_{1}}{{\phi }_{1}}\hat{H}{{c}_{1}}{{\phi }_{1}}+{{c}_{1}}{{\phi }_{1}}\hat{H}{{c}_{2}}{{\phi }_{2}}+{{c}_{2}}{{\phi }_{2}}\hat{H}{{c}_{1}}{{\phi }_{1}}+{{c}_{2}}{{\phi }_{2}}\hat{H}{{c}_{2}}{{\phi }_{2}}} \right)}}}}{{\int{{\left( {c_{1}^{2}\phi _{1}^{2}+2{{c}_{1}}{{c}_{2}}{{\phi }_{1}}{{\phi }_{2}}+c_{2}^{2}\phi _{2}^{2}} \right)d\tau }}}}---(4)\]

Now,

\[\displaystyle {{H}_{{11}}}=\int{{{{\phi }_{1}}\hat{H}}}{{\phi }_{1}}d\tau \]
\[\displaystyle {{H}_{{12}}}={{H}_{{21}}}=\int{{{{\phi }_{1}}\hat{H}}}{{\phi }_{2}}d\tau =\int{{{{\phi }_{2}}\hat{H}}}{{\phi }_{1}}d\tau \]
\[\displaystyle {{H}_{{22}}}=\int{{{{\phi }_{2}}\hat{H}}}{{\phi }_{2}}d\tau \]
\[\displaystyle {{S}_{{11}}}=\int{{\phi _{1}^{2}}}d\tau \]
\[\displaystyle {{S}_{{22}}}=\int{{\phi _{1}^{2}}}d\tau \]
\[\displaystyle {{S}_{{12}}}={{S}_{{21}}}=\int{{{{\phi }_{1}}{{\phi }_{2}}}}d\tau \]

Above notation, H11, H12, H21, S11, and S22 are matrix elements. Put this notation in equation (4) we can write as,

\[\displaystyle E=\frac{{c_{1}^{2}{{H}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{H}_{{12}}}+c_{2}^{2}{{H}_{{22}}}}}{{c_{1}^{2}{{S}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{S}_{{12}}}+c_{2}^{2}{{S}_{{22}}}}}---(5)\]

Energy is always obtained at the upper bound, and the ground state energy is obtained by variation theorem. Done by carrying out differentiation (dE/∂ci) differentiating concerning each coefficient (i=1,2) is derivative = 0

dE/∂ci =0, we get,

\[\displaystyle \frac{{\partial E}}{{\partial {{c}_{i}}}}=\frac{{\left( {c_{1}^{2}{{S}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{S}_{{12}}}+c_{2}^{2}{{S}_{{22}}}} \right)+\left( {2{{c}_{1}}{{H}_{{11}}}+2{{c}_{2}}{{H}_{{12}}}} \right)}}{{{{{\left( {c_{1}^{2}{{S}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{S}_{{12}}}+c_{2}^{2}{{S}_{{22}}}} \right)}}^{2}}}}-\frac{{\left( {c_{1}^{2}{{H}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{H}_{{12}}}+c_{2}^{2}{{H}_{{22}}}} \right)\left( {2{{c}_{1}}{{S}_{{11}}}-2{{c}_{2}}{{S}_{{12}}}} \right)}}{{{{{\left( {c_{1}^{2}{{S}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{S}_{{12}}}+c_{2}^{2}{{S}_{{22}}}} \right)}}^{2}}}}=0---(6)\]

Here,

\[\displaystyle {{c}_{1}}{{H}_{{11}}}+{{c}_{2}}{{H}_{{12}}}=\frac{{\left( {c_{1}^{2}{{H}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{H}_{{12}}}+c_{2}^{2}{{H}_{{22}}}} \right)\left( {{{c}_{1}}{{S}_{{11}}}+{{c}_{2}}{{S}_{{12}}}} \right)}}{{c_{1}^{2}{{S}_{{11}}}+2{{c}_{1}}{{c}_{2}}{{S}_{{12}}}+c_{2}^{2}{{S}_{{22}}}}}---(7)\]

Equations (5) and (7) we get,

\[\displaystyle {{c}_{1}}{{H}_{{11}}}+{{c}_{2}}{{H}_{{12}}}=E\left( {{{c}_{1}}{{S}_{{11}}}+{{c}_{2}}{{S}_{{12}}}} \right)\]

Or

\[\displaystyle {{c}_{1}}\left( {{{H}_{{11}}}-E{{S}_{{11}}}} \right)+{{c}_{2}}\left( {{{H}_{{12}}}-E{{S}_{{12}}}} \right)=0----(8)\]

Same, minimizing E concerning c2 that dE/∂c2 =0, we get,

\[\displaystyle {{c}_{1}}\left( {{{H}_{{21}}}-E{{S}_{{21}}}} \right)+{{c}_{2}}\left( {{{H}_{{22}}}-E{{S}_{{22}}}} \right)=0----(9)\]

From equations (8) and (9), The system of two simultaneous linear equations in two unknowns, c1 and c2, In the principle of linear algebra, an essential condition for this equation has a non-trivial solution is determinant of the coefficient c1 and c2 should dematerialize.

\[\displaystyle \left| \begin{array}{l}{{H}_{{11}}}-E{{S}_{{11}}}{{H}_{{12}}}-E{{S}_{{12}}}\\{{H}_{{21}}}-E{{S}_{{21}}}{{H}_{{22}}}-E{{S}_{{22}}}\end{array} \right|=0----(10)\]

Equation (10) is the 2×2 secular equation, and the left-hand side of this equation is known as the secular determinant. MO of the form,

\[\displaystyle {{\psi }_{i}}=\sum\limits_{{j=1}}^{n}{{{{c}_{{ij}}}}}{{\phi }_{j}}\left( {i=1,2,3,......n} \right)----(11)\]

Here, Φs = Atomic orbitals obtain an n×n secular determinantal equation,

\[\displaystyle \left| \begin{array}{l}{{H}_{{11}}}-{{S}_{{11}}}E{{H}_{{12}}}-{{S}_{{12}}}E..........{{H}_{{1n}}}-{{S}_{{1n}}}E\\{{H}_{{21}}}-{{S}_{{21}}}E{{H}_{{22}}}-{{S}_{{22}}}E..........{{H}_{{2n}}}-{{S}_{{2n}}}E\\{{H}_{{31}}}-{{S}_{{31}}}E{{H}_{{32}}}-{{S}_{{32}}}E..........{{H}_{{3n}}}-{{S}_{{3n}}}E\\.............................................\\.............................................\\.{{H}_{{n1}}}-{{S}_{{n1}}}E{{H}_{{n2}}}-{{S}_{{n2}}}E.........{{H}_{{nn}}}-{{S}_{{nn}}}E\end{array} \right|=0-----(12)\]

Equation (12) gives the value of MOs of n-electron molecules, and equation (10) obtains MOs energies for diatomic molecules.

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About the author

Bhoomika Sheladiya

BSc. (CHEMISTRY) 2014- Gujarat University
MSc. (PHYSICAL CHEMISTRY) 2016 - School of Science, Gujarat University

Junior Research Fellow (JRF)- 2019
AD_HOC Assistant Professor-(July 2016 to November 2021)

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