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Quantum Chemistry

Ira N. Levine

Chapter 14

Theorems of Molecular Quantum Mechanics - all with Video Answers

Educators

Chapter Questions

03:15

Problem 1

$$
\begin{array}{c|c|c|c|c|c|c|c}
\hline \text { Sec. } & 14.1 & 14.2 & 14.3 & 14.4 & 14.5 & 14.6 & 14.7 \\
\hline \text { Probs. } & 14.1-14.2 & 14.3-14.5 & 14.6-14.22 & 14.23-14.31 & 14.32-14.36 & 14.37-14.39 & 14.40-14.41 \\
\hline
\end{array}
$$
Derive (14.6) for $\rho_{\mathrm{VB}}$ and $\rho_{\mathrm{MO}}$ -

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:56

Problem 2

Show that $\rho_{\mathrm{MO}}$ of (14.6) is greater than $\rho_{\mathrm{VB}}$ of (14.6) at the midpoint of the line joining the nuclei.

Lottie Adams
Lottie Adams
Numerade Educator

Problem 3

Show that the dipole moment (14.9) of a system of charges is independent of the choice of the coordinate origin, provided the system has no net charge.

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Problem 4

(a) Explain why the permanent dipole moment of a many-electron atom in a stationary state is always zero. (b) Explain why the permanent electric dipole moment of H can be nonzero for certain excited states. (c) Show qualitatively that two of the four correct zeroth-order functions of Prob. 9.23 give nonzero permanent dipole moments.

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05:26

Problem 5

For $\mathrm{NaCl}, R_c=2.36 \AA$. The ionization energy of Na is 5.14 eV , and the electron affinity of Cl is 3.61 eV . Use the simple model of NaCl as a pair of spherical ions in contact to estimate $D_c$ and the dipole moment of NaCl . Compare with the experimental values $D_c=4.25 \mathrm{eV}$ and $\mu=9.0 \mathrm{D}$. [One debye (D) is $3.33564 \times 10^{-30} \mathrm{C} \mathrm{m}$.]

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator

Problem 6

Prove that the one-electron Hartree-Fock operator (14.26) is Hermitian.

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03:28

Problem 7

Explain the origin of the extra terms in the molecular Hartree-Fock operator (14.26) as compared with the atomic Hartree operator of (11.9) and (11.7).

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 8

Verify that the Coulomb and exchange integrals $J_{i j}$ and $K_{i j}$ can be written in terms of the Coulomb and exchange operators of Section 14.3 as

$$
J_{i j}=\left\langle\phi_i(1)\right| \hat{J}_j(1)\left|\phi_i(1)\right\rangle, \quad K_{i j}=\left\langle\phi_i(1)\right| \hat{K}_j(1)\left|\phi_i(1)\right\rangle
$$

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Problem 9

Verify Eq. (14.40) for the $\hat{K}_j$ integral over basis functions.

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Problem 10

(a) Use the definition (14.42) of $P_{\mathrm{nr}}$ to show that $P_{r s}=P_{s r}^*$, meaning that the density matrix $\mathbf{P}$ is a Hermitian matrix. (b) Show that Eq. (14.45) can be written as $E_{\mathrm{HF}}=\frac{1}{2} \operatorname{Tr}\left(\mathbf{P}^* \mathbf{F}+\mathbf{P}^* \mathbf{H}^{\text {cone }}\right)+V_{N N}$, where $\operatorname{Tr}$ denotes the trace of a matrix (Section 7.10) and the $\mathbf{P}, \mathbf{F}$, and $\mathbf{H}^{\text {oce }}$ matrices have elements $P_{r r}, F_{r z}$, and $H_{r z}^{\text {pore }}$. (c) Verify that (14.42) can be written as $\mathbf{P}^*=2 \mathbf{C C}^{\dagger}$, where $\mathbf{C}$ is the matrix of coefficients $c_{\text {w }}$,

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Problem 11

From (14.5), show that $\int_{-x}^x \int_{-x}^x \int_{-x}^x \rho d x d y d z=n$, where $\rho$ is the electron probability density of an $n$-electron molecule. (b) Use the result of (a) and Eq. (14.43) to show that $n=\Sigma, \Sigma_s P_{r s} S_{r n}=\Sigma, \Sigma_s P_{r n} S_{r r}{ }^*$. (c) Show that $n=\operatorname{Tr}\left(\mathbf{P S}^*\right)$, which becomes $n=\operatorname{Tr}(\mathbf{P S})$ for real basis functions. Here, $\mathbf{P}$ and $\mathbf{S}$ are the density and overlap matrices.

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05:41

Problem 12

Verify the equations for $H_{11}^{\text {coee }}$ and $H_{22}^{\text {cove }}$ in the Section 14.3 example.

Keshav Singh
Keshav Singh
Numerade Educator

Problem 13

Verify the equalities ( 14.47 ) for electron-repulsion integrals.

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Problem 14

Use Eq. (9.124) of Prob. 9.14 to verify the expressions for the integrals $(11 \mid 22)$ and (22|22) in the Section 14.3 example.

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01:30

Problem 15

For the He-atom SCF calculation in Section 14.3, find the initial estimate of $c_{11} / c_{21}$ given by the approximation $F_{r s}=H_{r s}^{\text {eose }}$.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 16

(a) Verify the equations for $F_{11}, F_{12}$, and $F_{22}$ that immediately precede Eq. (14.50). (b) Verify Eqs. (14.50) to (14.52) for $F_{11}, F_{12}$, and $F_{22}$.

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02:02

Problem 17

Derive (14.48) from the normalization condition for $\phi_1$.

James Kiss
James Kiss
Numerade Educator

Problem 18

Verify the numerical results for $P_{11}, P_{12}, P_{22}, F_{11}, F_{12}, F_{22}, \varepsilon_1, \varepsilon_2, c_{11}$, and $c_{21}$ obtained on the last cycle of calculation in the Section 14.3 example [Eqs. (14.53), (14.54), and the preceding and following equations].

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Problem 19

Repeat the He SCF calculation of Section 14.3 using the same basis functions but starting with the initial guess $c_{11}=c_{21}$ and $c_{21}$ determined by the normalization condition (14.48).

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Problem 20

(a) Write a computer program that will perform the helium-atom SCF calculation of Section 14.3. Have the input to the program be $\zeta_1, \zeta_2$, and the initial guess for $c_{11} / c_{21}$. Do not use the Section 14.3 values of the integrals, but have the program calculate all integrals from $\zeta_1$ and $\zeta_2$. Have the program print $c_{11}, c_{21}, \varepsilon_1$, and $\varepsilon_2$ for each cycle of calculation. Use the convergence criterion that $c_{11}$ and $c_{21}$ each differ from the $c_{11}$ and $c_{21}$ values of the previous cycle by less than $10^{-5}$ atomic units. (b) Use your program with $\zeta_1=1.45$ and $\zeta_2=2.91$ to find the number of iterations needed to reach convergence for each of these initial guesses for $c_{11} / c_{21}: 100,10,1,0,-1,-10,-100$. (c) Run the program with $\zeta_1=1.45363$ and $\zeta_2=2.91093$ to verify the SCF energy given for these $\zeta$ 's at the end of the Section 14.3 example. (d) Run the program with $\zeta_1$ changed by +0.01 and by -0.01 from the value in (c), and verify that each $E_{\mathrm{HF}}$ obtained is higher than that in (c). Repeat for $\zeta_2$.

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Problem 21

Calculate $\rho$ for the He SCF wave function of the Section 14.3 example at $r=0$ and at $r=1$ bohr.

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Problem 22

Given that $\phi_i=\sum_s c_{s i}^{\prime} \chi_s^{\prime}$, where the $\chi_s^{\prime}$ functions are orthonormal, show that the orbitals $\phi_i$ form an orthonormal set if the matrix $\mathbf{C}^{\prime}$ of coefficients $c_{s i}^{\prime}$ is unitary.

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00:16

Problem 23

Which of the following functions are homogeneous? Give the degree of homogeneity. (a) $x+3 y z$; (b) 179 ; (c) $x^2 / y z^3$; (d) $\left(a x^3+b x y^2\right)^{1 / 2}$.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
02:37

Problem 24

Let 1 and 2 be two bound stationary states of an atom, with $E_2>E_1$. For which state is the average electronic kinetic energy larger?

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator

Problem 25

Show that the hypervirial theorem follows from Eq. (7.113).

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20:07

Problem 26

(a) Calculate $\langle T\rangle$ and $\langle V\rangle$ for the helium-atom trial function (9.56). All the needed integrals were evaluated in Chapter 9 . (b) Verify that the virial theorem is satisfied for $\zeta=Z-5 / 16$ but not for $\zeta=\mathrm{Z}$.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:55

Problem 27

A one-dimensional harmonic oscillator in a stationary state has $\langle T\rangle=5.0 \times 10^{-19} \mathrm{~J}$. Find $E$ and $\langle V\rangle$ for this state.

Anand Jangid
Anand Jangid
Numerade Educator
01:22

Problem 28

A certain excited stationary state of He has an energy of -59.10 eV [with the zero of energy chosen as in Eq. (11.1)]. Find $\langle T\rangle$ for this state. (Assume relativistic effects are negligible.)

Narayan Hari
Narayan Hari
Numerade Educator
20:07

Problem 29

A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is 10 eV , calculate $\langle T\rangle$ and $\langle V\rangle$ for the ground state.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
14:53

Problem 30

The zero level of potential energy is arbitrary and we can always add a constant $C$ to the potential energy function $V$. (See Prob. 4.52.) If we add $C$ to $V$, state what happens to each of the following for a stationary state: $\langle V\rangle,\langle T\rangle, E$. Do these results contradict the virial theorem? Explain your answer.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
08:29

Problem 31

Prove that for a bound stationary state: (a) $\left\langle p_x\right\rangle=0$; (b) $\langle\partial V / \partial x\rangle=0$.

Keshav Singh
Keshav Singh
Numerade Educator
14:41

Problem 32

The $U(R)$ curve for a diatomic-molecule repulsive electronic state can be roughly approximated by the function $a e^{-\Delta R}-c$, where $a, b$, and $c$ are positive constants with $a>c$. (This function omits the van der Waals minimum and fails to go to infinity at $R=0$.) Sketch $U,\left\langle T_{\mathrm{el}}\right\rangle$, and $\langle V\rangle$ as functions of $R$ for this function.

Bret Rosen
Bret Rosen
Numerade Educator

Problem 33

Prove that $\partial\langle V\rangle / \partial R$ must be nonnegative at $R=R_z$; that is, $\langle V\rangle$ cannot be increasing with decreasing $R$ as we go through the minimum in the $U(R)$ curve (Fig. 14.1). State and prove the corresponding theorem for $\left\langle T_{\mathrm{el}}\right\rangle$.

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Problem 34

Let $\psi$ be the complete wave function for a molecule, with the Born-Oppenheimer approximation $\psi=\psi_{\text {el }} \psi_N$ not necessarily holding. Is it true that

$$
2\langle\psi| \hat{T}_{\mathrm{el}}+\hat{T}_N|\psi\rangle=-\langle\psi| \hat{V}|\psi\rangle
$$

where $\hat{T}_{\text {el }}$ and $\hat{T}_N$ are the kinetic-energy operators for the electrons and nuclei and $\hat{V}$ is the complete potential-energy operator? Justify your answer.

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02:32

Problem 35

Given that $D_c=4.75 \mathrm{eV}$ and $R_c=0.741 \AA$ for the ground electronic state of $\mathrm{H}_2$, find $U\left(R_e\right),\left.\langle V\rangle\right|_{R_z},\left.\left\langle V_{\mathrm{el}}\right\rangle\right|_{R_z}$, and $\left.\left\langle T_{\mathrm{el}}\right\rangle\right|_{R_z}$ for this state.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:14

Problem 36

The Fues potential-energy function for nuclear vibration of a diatomic molecule is $U(R)=U(\infty)+D_e\left(-2 R_e / R+R_e^2 / R^2\right)$. Find the expressions for $\left\langle T_{\mathrm{el}}\right\rangle$ and $\langle V\rangle$ predicted by this potential and comment on the results.

Hunza Gilgit
Hunza Gilgit
Numerade Educator

Problem 37

(a) Apply the generalized Hellmann-Feynman theorem with $Z$ as the parameter to find $\langle 1 / r\rangle$ for the hydrogenlike-atom bound states $\psi_{\text {nion }}$ (b) Since hydrogenlike functions $\psi_{\text {nion }}$ with the same $n$ but different $l$ or $m$ have the same energy, we must be sure that the functions $\psi_{\text {mon }}$ are the correct zeroth-order functions for the perturbation of varying Z. Use a theorem of Section 9.6 to verify this.

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Problem 38

Use the generalized Hellmann-Feynman theorem to find $\left\langle p_x^2\right\rangle$ for the one-dimensional harmonic-oscillator stationary states. Check that the result obtained agrees with the virial theorem.

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Problem 39

Differentiate (9.7) and (9.14) with respect to $\lambda$, substitute the results into the generalized Hellmann-Feynman theorem, and let $\lambda=0$ to derive the perturbation-theory equation $E_n^{(1)}=\left\langle\psi_n^{(0)}\right| \hat{H}^{\prime}\left|\psi_n^{(0)}\right\rangle$.

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Problem 40

Use $F_{2, \Delta}=-\partial U / \partial z_a$ and (14.85) to show that in Fig. 14.4, $F_{2, \Delta}=-(\partial U / \partial R)\left[\left(z_a-z_b\right) / R\right]$. Find a similar equation for $F_{2, b}$ and verify that $F_{2, a}=-F_{2, b}$

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02:43

Problem 41

The $R_c$ values for the ground electronic states of $\mathrm{HF}, \mathrm{HCl}, \mathrm{HBr}$, and HI are $0.92,1.27,1.41$, and $1.61 \AA$. The surface enclosing the antibinding region "behind" the proton in these molecules intersects the internuclear axis at two points, one of which is the proton location. Calculate the distance between these two points of intersection for each of the hydrogen halides.

Dading Chen
Dading Chen
Numerade Educator