19. Atomic orbitals

Schrödinger equation: wavefunction, electron density, concept of atomic orbitals, quantum numbers, s and p orbitals, shape, nodal surfaces, energy, degeneracy

Schrödinger equation

·    

o     : Hamilton operator

o      : wavefunction

o   E:energy

Wavefunction():

·       mathematical function that describes the quantum state of a particle or system of particles

Electron density (²):

·       the probability of finding an electron around a specific location around the atomic nucleus

Concept of atomic orbitals

·       regions where the electron density is high (where the electron is present 90% of the time)

Quantum numbers

·       Main quantum number (n)

o   the main quantum number determines the primary energy level and size of the orbital. Higher values of n correspond to orbitals that are further away from the nucleus and have higher energy

o   values: n=1,2,3….

·       Orbital quantum number (L)

o   the orbital quantum number determines the shape of the orbital and contributes to the orbital’s energy in multi-electron atoms

o   values: l=0,1,2,3…..n-1

§  L=0 → s orbitals (spherical)

§  L=1 → p orbitals (dumbbell-shaped)

§  L=2 → d orbitals (cloverleaf-shaped)

§  L=3 → f orbitals (complex shapes)

·       Magnetic quantum number (m_L)

o   the magnetic quantum number determines the orientation of the orbital in space relative to other orbitals

o   values: (m_L)= -L,…,0,…+L

·       Spin quantum number (m_s)

o   the spin quantum number specifies the orientation of the electron’s spin

o   values: m_s= -½ , ½

s and p orbitals

_·               s orbitals:

_o      spherical shape

_o      no nodal surfaces

_o      1 s orbital for every value of n

_o      the electron density highest at the nucleus and decreases with distance)

_o      L=0

_·               p orbitals:

_o      dumbbell-shaped

_o      one nodal surface

_o      3 p orbitals for every value of n

_o      n≥2, L=1

Nodal surfaces

·      regions  where the electron density(²) is 0

o   radial nodes (sphere shaped nodal surfaces)

o   angular nodes (plane shaped nodal surfaces)

Energy levels

·       the energy of an orbital depends on the n and in multi-electron atoms (basically anything other than hydrogen) on L too

·       electrons in higher main orbitals have higher energy

Degeneracy

·       refers to the situation when two or more quantum states share the same energy level

·       there are multiple orbitals with the same energy level

·       for example in the case of n=2 we have 4 possible orbitals 2s 2p_x 2p_y 2p_z, all three p orbitals are on the same energy level, while s has a lower energy level than that of the p orbitals → the 3 p orbitals are degenerate

(note: L is supposed to be non-capital but it literally looks like a one)