5. (McQuarrie 9-25) Show that the atomic determinantal wave function, 1 V(r1, r2) = 1sa (1) 1sẞ(1) √2 1sa(2) 1sẞ(2) is normalized when the 1s orbitals are normalized. Hint: Don't freak out about this problem-it's easier than it looks! Expand the Slater determinant to get the wavefunction in a more usable form, then set up the normalization integral for . Multiply everything out, and you should have complicated-looking integrals such as this: ( 1sa(1) 1sẞ(2) | 1sa(1) 1sẞ(2) ) While such integrals might look scary, they're actually straightforward. An integral like the one above has four pieces: integration over (a) the spatial coordinates of electron 1, (b) the spatial coordinates of electron 2, (c) the spin coordinates of electron 1, and (d) the spin coordinates of electron 2. You can break up these multi-dimensional integrals into separate integrals for each piece. For example: ( 1sa(1) 1sẞ(2) | 1sa(1) 1sß(2) ) = (1s(1)|1s(1)) · (a(1)|a(1)) · (1s(2)|1s(2)) · (ß(2)|ß(2)) Then you can readily evaluate each individual integral on the right-hand side using standard orthonormality relations. In the example above, each of the four integrals on the right-hand- side equals 1 by normalization. All of the integrals in this problem can be evaluated similarly, without ever having to do any messy calculus.
5. (McQuarrie 9-25) Show that the atomic determinantal wave function, 1 V(r1, r2) = 1sa (1) 1sẞ(1) √2 1sa(2) 1sẞ(2) is normalized when the 1s orbitals are normalized. Hint: Don't freak out about this problem-it's easier than it looks! Expand the Slater determinant to get the wavefunction in a more usable form, then set up the normalization integral for . Multiply everything out, and you should have complicated-looking integrals such as this: ( 1sa(1) 1sẞ(2) | 1sa(1) 1sẞ(2) ) While such integrals might look scary, they're actually straightforward. An integral like the one above has four pieces: integration over (a) the spatial coordinates of electron 1, (b) the spatial coordinates of electron 2, (c) the spin coordinates of electron 1, and (d) the spin coordinates of electron 2. You can break up these multi-dimensional integrals into separate integrals for each piece. For example: ( 1sa(1) 1sẞ(2) | 1sa(1) 1sß(2) ) = (1s(1)|1s(1)) · (a(1)|a(1)) · (1s(2)|1s(2)) · (ß(2)|ß(2)) Then you can readily evaluate each individual integral on the right-hand side using standard orthonormality relations. In the example above, each of the four integrals on the right-hand- side equals 1 by normalization. All of the integrals in this problem can be evaluated similarly, without ever having to do any messy calculus.
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Transcribed Image Text:5. (McQuarrie 9-25) Show that the atomic determinantal wave function,
1
V(r1, r2)
=
1sa (1) 1sẞ(1)
√2 1sa(2) 1sẞ(2)
is normalized when the 1s orbitals are normalized.
Hint: Don't freak out about this problem-it's easier than it looks! Expand the Slater
determinant to get the wavefunction in a more usable form, then set up the normalization
integral for . Multiply everything out, and you should have complicated-looking integrals
such as this:
( 1sa(1) 1sẞ(2) | 1sa(1) 1sẞ(2) )
While such integrals might look scary, they're actually straightforward. An integral like the
one above has four pieces: integration over (a) the spatial coordinates of electron 1, (b) the
spatial coordinates of electron 2, (c) the spin coordinates of electron 1, and (d) the spin
coordinates of electron 2. You can break up these multi-dimensional integrals into separate
integrals for each piece. For example:
( 1sa(1) 1sẞ(2) | 1sa(1) 1sß(2) ) = (1s(1)|1s(1)) · (a(1)|a(1)) · (1s(2)|1s(2)) · (ß(2)|ß(2))
Then you can readily evaluate each individual integral on the right-hand side using standard
orthonormality relations. In the example above, each of the four integrals on the right-hand-
side equals 1 by normalization. All of the integrals in this problem can be evaluated similarly,
without ever having to do any messy calculus.
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