Atomic orbital & Hydrogen-atom wave function Derivation of hydrogen-atom wave Schrdinger eq. of H atom Derivation of hydrogen-atom wave function E = - 2n 2 1 Energy Solution n,l,m (r, ,) R n,l (r) Y l,m ( ,), = x Hydrogen-atom wave function Radial wave function Spherical Harmonics n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l realized Hydrogen-atom wave function n,l,m (r, ,) real R n,l (r) Y l,m ( ,) x= Realized Spherical Harmonics eliminate a imaginary by linear combination separate the variables of ( r, , ) n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l where convert from the xyz to the r (polar) coordinate xyz convert to the atomic unit Schrdinger eq. of H atom Load to the Atomic Orbitals n,l,m (r, ,) real R n,l (r) x = n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l where Y 0,0 Y 1,0 Y + 1,1 Y - 1,1 Y 2,0 Y + 2,1 Y + 2,2 Y - 2,1 Y - 2,2 Y l,m ( ,) n=1 n=2 n=3 R 1,0 R 2,0 R 2,1 r R 3,0 R 3,1 R 3,2 l=0 l=1 l=2 2 n=1 n=2 n=3 R 1,0 R 2,0 R 2,1 r R 3,0 R 3,1 R 3,2 Load to the Atomic Orbitals n,l,m (r, ,) real R n,l (r) x= n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l where Y 0,0 Y 1,0 Y + 1,1 Y - 1,1 Y 2,0 Y + 2,1 Y + 2,2 Y - 2,1 Y - 2,2 Y l,m ( ,) 2 3D representation l=2 : d z 2 orbital d orbital z2z2 This transformation is similar to that of the world map. conversion Y 2,0 (r,x,y,z) Load to the Atomic Orbitals n,l,m (r, ,) real R n,l (r) x= n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l where Y 0,0 Y 1,0 Y + 1,1 Y - 1,1 Y 2,0 Y + 2,1 Y + 2,2 Y - 2,1 Y - 2,2 Y l,m ( ,) n=1 n=2 n=3 R 1,0 R 2,0 R 2,1 r R 3,0 R 3,1 R 3,2 n=1 n=2 n=3 R 1,0 R 2,0 R 2,1 r R 3,0 R 3,1 R 3,2 dz2dz2 d zx d yz d xy d x 2 -y 2 pzpz pxpx pypy s n,l,m (r, ,) real x= Y l,m ( ,) Load to the Atomic Orbitals Y l,m (r,x,y,z) R n,l (r) R 1,0 R 2,0 R 3,0 n=1 n=2 n=3 s s orbital 2s 3s x = 1s x R n,l (r) n,l,m (r,x,y,z) real = Y l,m (r,x,y,z) R 2,1 R 3,1 n=2 n=3 pzpz pxpx pypy p orbital x l=1 = 3p z 3p x 3p y 2p z 2p x 2p y x R n,l (r) n,l,m (r,x,y,z) real = Y l,m (r,x,y,z) dz2dz2 d zx d yz d xy d x 2 -y 2 d orbital x = 3d z 2 3d zx 3d yz 3d xy 3d x 2 -y 2 l=2 R 3,2 n=3 x R n,l (r) n,l,m (r,x,y,z) real = Y l,m (r,x,y,z) Atomic Orbitals and Energies of the Hydrogen Atom n=1 n=2 n=3 n=4 E = - 2n 2 1 ( K shell ) ( L shell ) ( M shell ) ( N shell ) l=1l=2l=3l=0 m=1m=2m=3 m=0 m=1 m=2 m=0 m=1 m=0 n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m = 0, 1, 2,..., l principal quantum number orbital angular momentum quantum number magnetic quantum number Atomic Orbitals and Energies of the Hydrogen Atom n,l,m (r, ,) n=1 n=2 n=3 n=4 E = - 2n 2 1 Atomic Orbitals and Energies of the Hydrogen Atom 4p z 4p x 4p y 4d z 2 4d zx 4d yz 4d xy 4d x 2 -y 2 4s 4f x(5z 2 -r 2 ) 4f y(5z 2 -r 2 ) 4f z(x 2 -y 2 ) 4f xyz 4f x(x 2 -3y 2 ) 4f y(3x 2 -y 2 ) 2s 3s 1s 3p z 3p x 3p y 2p z 2p x 2p y 3d z 2 3d zx 3d yz 3d xy 3d x 2 -y 2 4f z(5z 2 -3r 2 ) n,l,m (r,x,y,z) real n,l,m (r, ,) n=1 n=2 n=3 n=4 E = - 2n 2 1 Atomic Orbitals and Energies of the Hydrogen Atom 4p z 4p x 4p y 4d z 2 4d zx 4d yz 4d xy 4d x 2 -y 2 4s 4f x(5z 2 -r 2 ) 4f y(5z 2 -r 2 ) 4f z(x 2 -y 2 ) 4f xyz 4f x(x 2 -3y 2 ) 4f y(3x 2 -y 2 ) 2s 3s 1s 3p z 3p x 3p y 2p z 2p x 2p y 3d z 2 3d zx 3d yz 3d xy 3d x 2 -y 2 4f z(5z 2 -3r 2 ) n,l,m (r,x,y,z) real 4p z 4p x 4p y 4d z 2 4d zx 4d yz 4d xy 4d x 2 -y 2 4s 4f x(5z 2 -r 2 ) 4f y(5z 2 -r 2 ) 4f z(x 2 -y 2 ) 4f xyz 4f x(x 2 -3y 2 ) 4f y(3x 2 -y 2 ) 4f z(5z 2 -3r 2 ) 4f x(5z 2 -r 2 ) 4f y(5z 2 -r 2 ) 4f z(x 2 -y 2 ) 4f xyz 4f x(x 2 -3y 2 ) 4f y(3x 2 -y 2 ) 4f z(5z 2 -3r 2 ) 4d z 2 4d zx 4d yz 4d xy 4d x 2 -y 2 4p z 4p x 4p y 4s 3d z 2 3d zx 3d yz 3d xy 3d x 2 -y 2 2s 3s 1s 3p z 3p x 3p y 2p z 2p x 2p y