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GATE 2022 CY – Question 20

Physical Chemistry · Molecular Structure and Chemical Bonding: Born-Oppenheimer approximation; Valence bond theory treatment for hydrogen molecule and linear combination of atomic orbitals – molecular orbitals (LCAO-MO) theory. Concept of hybridization and hybrid orbitals. Applications of LCAO-MO theory to H2+, H2; Molecular orbital theory (MOT) of homo- and heteronuclear diatomic molecules. Hückel theory, applications to simple molecules, small conjugated π-electron systems. · 1 mark · Multiple choice

$1s_A$ and $1s_B$ are normalized eigenfunctions of hydrogen atoms $H_A,H_B$. If $S=\langle1s_A\mid1s_B\rangle$, which option is ALWAYS correct?

  1. $S=1$
  2. $S=0$
  3. S is an imaginary constant
  4. $0\le S\le1$

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Correct answer: (D) $0\le S\le1$

Explanation

Cauchy–Schwarz gives|S|≤1 for normalized functions. Conventional real hydrogen 1s orbitals are nonnegative, so their overlap is also nonnegative.

Hence **0≤S≤1, D**; it approaches zero at large separation and one for coincident identical orbitals.