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State the normalization condition for radial part of the atomic-orbital ?

State the normalization condition for radial part of the atomic-orbital ?

Chemistry
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
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Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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what is State the normalization condition for radial part of the atomic-orbital ?

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TABLE 6.4 The first few associated Laguerre polynomials. n = 1, n = 2, n = 3, n = 4, 1=0 1=0 1 = 1 1=0 1 = 1 1=2 1=0 1 = 1 1 = 2 1=3 L(x) = -1 L₂(x) = -2!(2-x) L3(x) = -3! L₁(x) = -3!(3 - 3x + 1x²) L(x)=-4!(4x) L(x) = -5! L₁(x) = −4!(4 - 6x + 2x² − £x³) L3(x) = -5!(10-5x + x²) L(x) = -6!(6-x) L'(x) = -7! The functions given by Equation 6.47 may look complicated, but notice that each one is just a polynomial multiplied by an exponential. The combinatorial factor in front assures that the R₁ (r) are normalized with respect to an integration over r, or that the R₁(r) satisfy R² (r) R (r) r²dr = 1 Note that the volume element here is rdr, which is the "r" part of the spherical coordinate volume element r² sin @drdodo. The complete hydrogen atomic wave functions are (6.48) (6.49) nim (r, 0, 0) = R₁ (r) Y™ (0, 0) The first few hydrogen atomic wave functions are given in Table 6.5. The normalization condition for hydrogen atomic wave functions is [drr² de sino dom (r, 0, 0)nim (r, 0, 0) = 1 Because Ĥ is Hermitian (Section 4-5), the functions must also be orthogonal. This orthogonality relationship is given by (6.50) f" drr² ["* de sine f S™ dom (r, 0, 0)nim (r, 0, 0) = Enn Summ (6.51) where the 8's are Krönecker deltas (Equation 4.30). 207
Transcribed Image Text:TABLE 6.4 The first few associated Laguerre polynomials. n = 1, n = 2, n = 3, n = 4, 1=0 1=0 1 = 1 1=0 1 = 1 1=2 1=0 1 = 1 1 = 2 1=3 L(x) = -1 L₂(x) = -2!(2-x) L3(x) = -3! L₁(x) = -3!(3 - 3x + 1x²) L(x)=-4!(4x) L(x) = -5! L₁(x) = −4!(4 - 6x + 2x² − £x³) L3(x) = -5!(10-5x + x²) L(x) = -6!(6-x) L'(x) = -7! The functions given by Equation 6.47 may look complicated, but notice that each one is just a polynomial multiplied by an exponential. The combinatorial factor in front assures that the R₁ (r) are normalized with respect to an integration over r, or that the R₁(r) satisfy R² (r) R (r) r²dr = 1 Note that the volume element here is rdr, which is the "r" part of the spherical coordinate volume element r² sin @drdodo. The complete hydrogen atomic wave functions are (6.48) (6.49) nim (r, 0, 0) = R₁ (r) Y™ (0, 0) The first few hydrogen atomic wave functions are given in Table 6.5. The normalization condition for hydrogen atomic wave functions is [drr² de sino dom (r, 0, 0)nim (r, 0, 0) = 1 Because Ĥ is Hermitian (Section 4-5), the functions must also be orthogonal. This orthogonality relationship is given by (6.50) f" drr² ["* de sine f S™ dom (r, 0, 0)nim (r, 0, 0) = Enn Summ (6.51) where the 8's are Krönecker deltas (Equation 4.30). 207
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