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Chapter 6: Problem 18

Consider the hypothetical linear \(\mathrm{H}_{3}\) molecule. The wavefunctions may be modelled by expressing them as \(\psi=c_{\Lambda} s_{A}+c_{B} s_{B}+c_{C} s_{C}\) the \(s_{i}\) denoting hydrogen 1 s-orbitals of the relevant atom. Use the Rayleigh-Ritz method to find the optimum values of the coefficients and the energies of the orbital. Make the approximations \(H_{\mathrm{ss}}=\alpha, H_{\mathrm{ss}^{\prime}}=\beta\) for neighbours but 0 for non-neighbours, \(S_{\mathrm{ss}}=1,\) and \(S_{\mathrm{ss}^{\prime}}=0\) Hint. Although the basis can be used as it stands, it leads to a \(3 \times 3\) determinant and hence to a cubic equation for the energies. A better procedure is to set up symmetry-adapted combinations, and then to use the vanishing of \(H_{i j}\) unless \(\Gamma^{(i)}=\Gamma^{(j)}\).

Short Answer

Expert verified
To find the optimum values of coefficients and the energies of the orbital in the hypothetical linear H3 molecule, we construct an energy functional and differentiate it to get a system of equations. But before that, we form symmetry-adapted combinations of the wave function to simplify the Hamiltonian. On solving this system of equations, we get the optimum coefficients and the energy of the orbital.

Step by step solution

01

Construction of Energy Functional

The Rayleigh-Ritz method involves minimizing an energy functional to find the energies of the system. Here, the energy functional would be the expectation value of the Hamiltonian with the wavefunction. For our system, \n\n\[E = \langle H \rangle = \frac{\langle \Psi | H |\Psi \rangle}{\langle \Psi |\Psi \rangle}\] with \(\Psi = c_{A} s_{A}+c_{B} s_{B}+c_{C} s_{C}\). \n\n The numerator of this fraction involves calculation of expectation value of the Hamiltonian while the denominator involves normalization of the wavefunction.
02

Symmetry-adapted Combinations

The next step is to simplify the above expression by using symmetry-adapted combinations to shrink the Hamiltonian matrix from 3x3 to 1x1. \n\nWe can form two such combinations, each belonging to a unique symmetry species: \(\Psi_1 = s_{A} + s_{B} + s_{C}\) and \(\Psi_2 = s_{A} - s_{B}\). \n\nWith these combinations, the Hamiltonian matrix size reduces to 1x1 for \(\Psi_1\) and 2x2 for \(\Psi_2\).
03

Solving for Coefficients and Energies

To find the optimum coefficients (c_A,c_B,c_C), differentiate the energy functional we derived in step 1 with respect to each of the coefficients and set the resulting expressions equal to zero. \n\nPerforming this for both our symmetry-adapted combinations \(\Psi_1\) and \(\Psi_2\) will lead to a simpler 1x1 and 2x2 system of linear equations, which can be solved to find the unknowns c_A,c_B,c_C and the energy \(E\). \n\nOnce we get the optimum coefficients, find \(E\) by substituting these values into the energy functional.

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Most popular questions from this chapter

Calculate the first-order correction to the energy of a ground-state harmonic oscillator subject to an anharmonic potential of the form \(a x^{3}+b x^{4}\) where \(a\) and \(b\) are small (anharmonicity) constants. Consider the three cases in which the anharmonic perturbation is present (a) during bond expansion \((x \geq 0)\) and compression \((x \leq 0)\) (b) during expansion only, (c) during compression only.

A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1 s-orbital (but with \(Z=2\) ). The electronelectron interaction is regarded as a perturbation, and calculation gives $$\int \psi_{1 s}^{2}\left(r_{1}\right)\left(\frac{e^{2}}{4 \pi \varepsilon_{0} r_{12}}\right) \psi_{1 s}^{2}\left(r_{2}\right) \mathrm{d} \tau=\frac{5}{4}\left(\frac{e^{2}}{4 \pi \varepsilon_{0} a_{0}}\right)$$ (see Example 7.2 ). Estimate (a) the binding energy of helium, (b) its first ionization energy. Hint. Use eqn 6.15 with \(E_{1}=E_{2}=E_{1 \mathrm{s}} .\) Be careful not to count the electronelectron interaction energy twice.

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).

A hydrogen atom in a \(2 \mathrm{s}^{1}\) configuration passes into a region where it experiences an electric field in the \(z\) -direction for a time \(\tau .\) What is its electric dipole moment during its exposure and after it emerges? Hint. Use eqn 6.62 with \(\omega_{21}=0 ;\) the dipole moment is the expectation value of \(-e z\) \\[\text { use } \int \psi_{2 \mathrm{s}} z \psi_{2 \mathrm{p}} \mathrm{d} \tau=3 a_{0}\\]

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.
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