Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
Question
Book Icon
Chapter 12, Problem 12.45E
Interpretation Introduction

Interpretation:

The variation theory treatment of H using ϕ=ekr2 as an unnormalized trial function is to be performed and specific expression for k is to be stated.

Concept introduction:

According to the variation theory, the lower the energy of a system the better is the approximation. Variation theory is based on the fact that any test system has average energy equal to or greater than the ground state energy of that system. The advantage of the variation theory is that any wavefunction can be taken for any test system.

Expert Solution & Answer
Check Mark

Answer to Problem 12.45E

The variation theory treatment of H using ϕ=ekr2 as an unnormalized trial function has been performed and specific expression for k is e4μ218π3ε024.

Explanation of Solution

The expression for trial function can be written as given below.

Etrial=ϕ*H^ϕdτϕ*ϕdτ

Substitute the values in the above expression as follows.

Etrial = ϕ * H ^ ϕdτ ϕ * ϕdτ= e k r 2 ( 2 2μ ( 1 r 2 r ( r 2 r ) ) e 2 4π ε 0 r ) e k r 2 dτ e 2k r 2 dτ= 22μ0 e k r 2 ( 1 r 2 r ( r 2 e k r 2 r ))dr0 e 2 4π ε 0 re 2k r 2 dr0e2k r 2dr=22μ0ek r 2( 1 r 2 r( r 2 ( 2kr ) e k r 2 ))dr0e24πε0re2kr2dr0e2kr2dr

The above equation can be simplified as given below.

= 22μ0 e k r 2 ( 2k r 2 ( 3 r 2 e kr 2k r 4 e kr ))dr0 e 2 4π ε 0 re 2k r 2 dr0e2k r 2dr=2μ0e2k r 2(32k r 2)dre24πε00e 2k r 2 rdr0e2kr2dr=32kμ22k2μ0r2e2k r 2dre24πε00e 2k r 2 rdr0e2kr2dr=32kμ22kμ×( π 8 k 3 )1/212( π 2k)1/2( e 24π ε 0)0e2k r 2dr4πr20e2kr2dr4πr2

Solve the integral as follows.

=32kμ22kμ×(π8k3)1/212(π2k)1/2(e24πε0)(14k)(π8k3)1/2=32kμ22μe2πε0(2kπ)1/2

The value of k can be calculated by differentiating the above expression with respect to k as follows.

Etrial=32kμ22μe2πε0(2kπ)1/2Etrialk=32μ12k1/2e2πε0(2π)1/2=0k=2e44π3ε02×μ294=e4μ218π3ε024

Substitute this value in the expression for trial energy as given below.

Etrial=32kμ22μe2πε0(2kπ)1/2=32μ(e4μ218π3ε024)22μe2πε0(2π)1/2(e4μ218π3ε024)1/2=e4μ6π3ε02222μe2πε0(2π)1/2(e4μ218π3ε024)1/2

Conclusion

The variation theory treatment of H using ϕ=ekr2 as an unnormalized trial function has been performed and specific expression for k is e4μ218π3ε024.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Predict the products of this organic reaction: H3O+ + ? • Draw all the reasonable products in the drawing area below. If there are no products, because no reaction will occur, check the box under the drawing area. • Include both major and minor products, if some of the products will be more common than others. • Be sure to use wedge and dash bonds if you need to distinguish between enantiomers. No reaction. Click and drag to start drawing a structure. dm
I
Draw the anti-Markovnikov product of the hydration of this alkene. this problem. Note for advanced students: draw only one product, and don't worry about showing any stereochemistry. Drawing dash and wedge bonds has been disabled for esc esc ☐ Explanation Check F1 1 2 F2 # 3 F3 + $ 14 × 1. BH THE BH3 2. H O NaOH '2 2' Click and drag to start drawing a structure. F4 Q W E R A S D % 905 LL F5 F6 F7 © 2025 McGraw Hill LLC. All Rights Reserved. Terms of Use | Privacy Center | Accessibility < & 6 7 27 8 T Y U G H I F8 F9 F10 F11 F12 9 0 J K L P + // command option Z X C V B N M H H rol option command

Chapter 12 Solutions

Physical Chemistry

Ch. 12 - Draw a diagram analogous to Figure 11.15, but now...Ch. 12 - Are mathematical expressions for the following...Ch. 12 - Prob. 12.13ECh. 12 - Prob. 12.14ECh. 12 - a Assume that the electronic energy of Li was a...Ch. 12 - Spin orbitals are products of spatial and spin...Ch. 12 - If 1 and 2 are the individual wavefunctions for...Ch. 12 - Show that the correct behavior of a wavefunction...Ch. 12 - Prob. 12.19ECh. 12 - Why isnt the electron configuration of beryllium,...Ch. 12 - Prob. 12.21ECh. 12 - Write a Slater determinant for the lithide ion,...Ch. 12 - Why does the concept of antisymmetric...Ch. 12 - a Construct Slater determinant wavefunctions for...Ch. 12 - Prob. 12.25ECh. 12 - Prob. 12.26ECh. 12 - Prob. 12.27ECh. 12 - Suppose an electron had three possible values of...Ch. 12 - Using a periodic table or Table 12.1, find the...Ch. 12 - Write an acceptable electron configuration for...Ch. 12 - Prob. 12.31ECh. 12 - Prob. 12.32ECh. 12 - Prob. 12.33ECh. 12 - An anharmonic oscillator has the potential...Ch. 12 - Prob. 12.35ECh. 12 - In a particle-in-a-box having length a, the...Ch. 12 - Prob. 12.37ECh. 12 - Prob. 12.38ECh. 12 - Prob. 12.39ECh. 12 - The Stark effect is the change in energy of a...Ch. 12 - Prob. 12.41ECh. 12 - Prob. 12.42ECh. 12 - Prob. 12.43ECh. 12 - Show that a variation theory treatment of H using...Ch. 12 - Prob. 12.45ECh. 12 - Explain why assuming an effective nuclear charge,...Ch. 12 - Prob. 12.47ECh. 12 - Consider a real system. Assume that a real...Ch. 12 - Prob. 12.49ECh. 12 - Prob. 12.50ECh. 12 - Prob. 12.51ECh. 12 - Prob. 12.52ECh. 12 - State the Born-Oppenheimer approximation in words...Ch. 12 - Prob. 12.54ECh. 12 - Spectroscopy deals with differences in energy...Ch. 12 - Prob. 12.56ECh. 12 - What is the bond order for the lowest excited...Ch. 12 - The helium atom was defined as two electrons and a...Ch. 12 - Explain how we know that the first in equation...Ch. 12 - Prob. 12.60ECh. 12 - Prob. 12.61ECh. 12 - Use molecular orbital arguments to decide whether...Ch. 12 - Prob. 12.63ECh. 12 - Prob. 12.65ECh. 12 - Prob. 12.67ECh. 12 - Prob. 12.68E
Knowledge Booster
Background pattern image
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
  • Text book image
    Physical Chemistry
    Chemistry
    ISBN:9781133958437
    Author:Ball, David W. (david Warren), BAER, Tomas
    Publisher:Wadsworth Cengage Learning,
Text book image
Physical Chemistry
Chemistry
ISBN:9781133958437
Author:Ball, David W. (david Warren), BAER, Tomas
Publisher:Wadsworth Cengage Learning,