We present a solution for hydrogenic atom composed of a nucleus made of Z protons and a single orbiting electron with Coulomb attraction in between them.
“Hydrogenic Schrödinger Orbitals” Kamil Walczak Department of Chemistry and Physical Sciences, Pace University, 1 Pace Plaza, New York, NY 10038 INTRODUCTION
Polar part of Angular Wave Equation (AWE):
We present a solution for hydrogenic atom composed of a nucleus made of Z protons and a single orbiting electron with Coulomb attraction in between them. The energy eigenvalue problem for hydrogen atom belongs to exactly solvable problems in quantum mechanics with crucial applications in chemistry and molecular biology. Using spherical polar coordinates, we show that threedimensional (3D) Schrödinger equation for hydrogenic atom is separable into three one-dimensional (1D) equations which may be solved analytically. The polar and azimuthal components of Angular Wave Equation (AWE) determine the shape and the spatial orientation of individual orbitals, while the Radial Wave Equation (RWE) captures the actual size of those orbitals. We also visualize hydrogenic Schrödinger orbitals with n ≤ 4, discuss nomenclature used in atomic physics, and explain experimental method used to probe atomic orbitals – photoionization spectroscopy.
1 d d m sin() 2 m () ( 1)m () d sin () sin() d
Nonrelativistic Schrödinger equation for hydrogenic atom: 2 2 Ze nm (r, , ) E n nm (r, , ) k r 2
Laplacian operator in spherical polar coordinates: 2 1 2 1 1 2 r sin() 2 2 r r r sin() sin ()
The hydrogenic wave function in the form of a product:
nm (r, , ) R n (r) m () m () Normalization condition for hydrogenic wave function: 2
d d sin( ) dr r | ( r , , ) | 1 n m 2
0
0
2
Nomenclature: Atomic Physics
2
Normalized expression for polar wave function:
(2 1) ( | m |)! |m| m () P (cos()) with 2 ( | m |)!
0,1,..., n 1
(associated Legendre polynomials of the first kind)
Radial Wave Equation Radial Wave Equation (RWE) for distance-dependent wave function:
1 d 2 d ( 1) Ze r R ( r ) E k n 2 2 2 r dr dr r r 2
Hydrogenic Atom (Schrödinger)
Hydrogenic Schrödinger Orbitals (2)
2
R n (r )
Photoionization Spectroscopy Hydrogenic Schrödinger Orbitals (3)
Normalized expression for radial wave function:
(n 1)! 2 1 R n (r ) exp L n () 3 r 2n[(n )!] 2 3
2Z r na 0
n 1,2,3,...,
with
(associated Laguerre polynomials)
Allowed energies for electrons (discrete spectrum of bound states):
k e Z 2 2Z En c 2 2 2 2 n 2 n 2
4
2
2
with
1 137
Hydrogenic Schrödinger Orbitals (1)
CLOSING REMARKS Hydrogenic Schrödinger Orbitals (4)
Since the angular parts of the wave functions to a large extent are independent of the form of potential in Schrödinger equation, the shapes of the orbitals are approximately the same for all atoms.
0
Angular Wave Equation Azimuthal part of Angular Wave Equation (AWE):
d m () 2 m m () 2 d 2
Normalized expression for azimuthal wave function:
1 m () exp im 2
with
m 0,1,2,...,
Principal number n describes orbital’s size (the larger n, the larger orbital’s range), azimuthal number ℓ describes orbital’s shape, while magnetic number m describes orbital’s orientation is space. Hydrogenic Schrödinger orbitals constitute a starting point in the description of molecules, because molecular orbitals may be expressed as a Linear Combination of Atomic Orbitals (LCAO). With the exception of the ground state (n=1), the hydrogenic eigenstates are degenerated, because their energies depend only on the principal (main) quantum number n.
