Atomic structure, term symbols, coupling schemes and spectra of many‑electron systems

Atomic Structure of Many-Electron Systems

Understanding the atomic structure of many-electron systems is crucial because it forms the foundation for explaining their chemical behavior and spectral properties. Unlike single-electron atoms (like hydrogen), atoms with multiple electrons exhibit complex interactions that significantly influence their energy levels and transitions.

In a hydrogen atom, electrons occupy distinct orbitals described by quantum numbers (n, l, m_l, m_s). The energy of these electrons depends solely on the principal quantum number, n. However, in many-electron atoms, additional factors come into play:

• Electron-Electron Repulsion: Electrons repel each other. This repulsion affects the energies of the orbitals, making them higher than they would be in a single-electron atom.
• Electron-Electron Shielding: Inner-shell electrons shield outer-shell electrons from the full positive charge of the nucleus. This shielding effect reduces the effective nuclear charge (Z_eff) experienced by the outer electrons.
• Orbital Penetration: Electrons in different types of orbitals (s, p, d, f) have different probabilities of being found close to the nucleus. This "penetration" effect also influences the energy levels, causing orbitals with lower angular momentum (like s orbitals) to be lower in energy than those with higher angular momentum (like p or d orbitals) for the same principal quantum number n.

These factors lead to a splitting of energy levels for a given n, which is not observed in hydrogen. For instance, in a lithium atom (Z=3), the electron configuration is 1s²2s¹. The two 1s electrons shield the 2s electron from the nucleus. The 2s electron experiences a lower effective nuclear charge and has a higher energy than if it were in a 1s orbital, but lower than a hypothetical 2s orbital in a bare Li⁺³ ion.

The ordering of orbital energies in many-electron atoms follows the Aufbau principle, Hund's rule, and the Pauli exclusion principle. The general order of filling is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, etc., though exceptions exist, particularly for transition metals.

Term Symbols for Many-Electron Atoms

To describe the electronic states of many-electron atoms, we use term symbols. These symbols provide a concise way to represent the total orbital angular momentum, total spin angular momentum, and the total angular momentum of the electrons in a given electronic configuration.

The term symbol is written as ^2S+1L_J, where:

• 2S+1 is the spin multiplicity.
• L is the total orbital angular momentum quantum number.
• J is the total angular momentum quantum number.

Total Orbital Angular Momentum (L)

Each electron in an atom has an orbital angular momentum described by the quantum number l. For a single electron, l can take values 0, 1, 2, ..., up to n-1, represented by the letters s, p, d, f, respectively.

In a many-electron atom, the total orbital angular momentum L is the vector sum of the individual orbital angular momenta (l_i) of all electrons: L = Σ l_i

The magnitude of the total orbital angular momentum is given by ħ√{L(L+1)}, where L is the total orbital angular momentum quantum number. L can take integer values from |l₁ - l₂| to l₁ + l₂ (for two electrons) or from Σ|l_i| down to Σl_i in steps of 1 (for many electrons), considering the Pauli exclusion principle.

The value of L is represented by a letter code:

Total Spin Angular Momentum (S)

Each electron also has an intrinsic spin angular momentum described by the spin quantum number s = 1/2. The total spin angular momentum S is the vector sum of the individual spin angular momenta (s_i) of all electrons: S = Σ s_i

The magnitude of the total spin angular momentum is given by ħ√{S(S+1)}, where S is the total spin quantum number. For a system of N electrons, S can range from |Σs_i| to Σs_i in steps of 1. For electrons with s=1/2, if all spins are parallel, S = N/2; if all spins are antiparallel, S = 0 (if N is even) or S = 1/2 (if N is odd).

Spin Multiplicity (2S+1)

The spin multiplicity is determined by the total spin quantum number S. It indicates the number of possible orientations of the total spin vector in space.

• If S = 0, multiplicity = 2(0)+1 = 1 (Singlet state)
• If S = 1/2, multiplicity = 2(1/2)+1 = 2 (Doublet state)
• If S = 1, multiplicity = 2(1)+1 = 3 (Triplet state)
• If S = 3/2, multiplicity = 2(3/2)+1 = 4 (Quartet state)

The spin multiplicity is written as a superscript to the left of the L symbol.

Total Angular Momentum (J)

The total angular momentum J arises from the coupling of the total orbital angular momentum L and the total spin angular momentum S. J = L + S

The magnitude of the total angular momentum is given by ħ√{J(J+1)}, where J is the total angular momentum quantum number. The possible values of J range from |L - S| to L + S in integer steps.

The value of J is written as a subscript to the right of the L symbol.

Determining Term Symbols for Electron Configurations

To determine the term symbols for a given electronic configuration, we need to consider the possible combinations of L and S values, respecting the Pauli exclusion principle.

Example: Nitrogen atom (N), configuration 1s²2s²2p³

The 1s and 2s electrons are in filled subshells. Electrons in filled subshells do not contribute to the term symbol because their orbital and spin angular momenta cancel out due to pairing. Thus, we only need to consider the three electrons in the 2p subshell.

For a single p electron (l=1): l=1, s=1/2. L=1, S=1/2. Possible J values: |1 - 1/2| = 1/2 to 1 + 1/2 = 3/2. So J = 1/2, 3/2. Term symbols: ²P_1/2, ²P_3/2.

For three electrons in the 2p subshell (p³): The three p orbitals are p_x, p_y, p_z. The three electrons can be arranged in different ways. We need to find the combinations of L and S that are allowed.

Possible combinations of (m_l, m_s) for three electrons in p orbitals: Electron 1: (1, +1/2) Electron 2: (0, +1/2) Electron 3: (-1, +1/2) Here, Σl_i = 1 + 0 + (-1) = 0. So L=0. Here, Σs_i = 1/2 + 1/2 + 1/2 = 3/2. So S=3/2. Multiplicity = 2S+1 = 2(3/2)+1 = 4. Possible J values: |L - S| = |0 - 3/2| = 3/2 to L + S = 0 + 3/2 = 3/2. So J=3/2. This gives the term symbol ⁴S_3/2.

Another possible arrangement: Electron 1: (1, +1/2) Electron 2: (1, -1/2) Electron 3: (0, +1/2) Here, Σl_i = 1 + 1 + 0 = 2. L=2. Here, Σs_i = 1/2 - 1/2 + 1/2 = 1/2. S=1/2. Multiplicity = 2S+1 = 2(1/2)+1 = 2. Possible J values: |L - S| = |2 - 1/2| = 3/2 to L + S = 2 + 1/2 = 5/2. So J=3/2, 5/2. This gives term symbols ²D_3/2, ²D_5/2.

Another possible arrangement: Electron 1: (1, +1/2) Electron 2: (0, +1/2) Electron 3: (0, -1/2) Here, Σl_i = 1 + 0 + 0 = 1. L=1. Here, Σs_i = 1/2 + 1/2 - 1/2 = 1/2. S=1/2. Multiplicity = 2S+1 = 2. Possible J values: |L - S| = |1 - 1/2| = 1/2 to L + S = 1 + 1/2 = 3/2. So J=1/2, 3/2. This gives term symbols ²P_1/2, ²P_3/2.

The ground state term symbol is the one with the highest spin multiplicity, and for terms with the same multiplicity, it is the one with the lowest L value. For excited states, the order is reversed. Hund's rules are used to determine the ground state term symbol.

For p³ configuration, the ground state term is ⁴S_3/2.

Coupling Schemes

In atoms with more than one electron, the individual orbital and spin angular momenta of the electrons interact. The way these interactions are combined is described by coupling schemes. The two main coupling schemes are Russell-Saunders (or LS coupling) and j-j coupling.

LS Coupling (Russell-Saunders Coupling)

LS coupling is dominant in lighter atoms. In this scheme, the orbital angular momenta of all electrons are first coupled together to form a total orbital angular momentum L. Similarly, the spin angular momenta of all electrons are coupled to form a total spin angular momentum S. These two total angular momenta, L and S, are then coupled to form the total angular momentum J.

L = Σ l_i S = Σ s_i J = L + S

This is the scheme used to derive term symbols like ^2S+1L_J. LS coupling assumes that the spin-orbit coupling is weak compared to the electron-electron repulsion.

Example: Helium atom (He), configuration 1s¹2s¹

Electron 1: l₁=0, s₁=1/2 Electron 2: l₂=0, s₂=1/2

Total orbital angular momentum L = l₁ + l₂ = 0 + 0 = 0. (S term) Total spin angular momentum S = s₁ + s₂. Possible values: - Spins parallel: S = 1/2 + 1/2 = 1. Multiplicity = 2(1)+1 = 3 (Triplet). - Spins antiparallel: S = 1/2 - 1/2 = 0. Multiplicity = 2(0)+1 = 1 (Singlet).

Combining L=0 with S=1: J = L + S = 0 + 1 = 1. Term symbol: ³S₁.

Combining L=0 with S=0: J = L + S = 0 + 0 = 0. Term symbol: ¹S₀.

So, the configuration 1s¹2s¹ gives rise to ¹S and ³S terms. The ¹S state is the ground state because it has lower electron-electron repulsion (due to paired spins not contributing to spin-dependent interactions). The ³S state is higher in energy.

In the presence of spin-orbit coupling, the ³S₁ state splits into different J levels. However, for S terms (L=0), J is always equal to S, so there is no splitting.

j-j Coupling

j-j coupling is more appropriate for heavier atoms where spin-orbit coupling is significant. In this scheme, the orbital angular momentum (l_i) and spin angular momentum (s_i) of each individual electron are coupled first to form a total angular momentum j_i for each electron.

j_i = l_i + s_i

The possible values for j_i are |l_i - s_i| to l_i + s_i. For an electron with l=1 (p orbital), s=1/2, j_i can be |1-1/2|=1/2 or 1+1/2=3/2.

These individual total angular momenta j_i are then coupled together to form the total angular momentum J for the atom.

J = Σ j_i

j-j coupling leads to a different set of possible energy levels compared to LS coupling. For example, for two electrons in p orbitals (p²), LS coupling gives ¹S, ³P, ¹D terms. In j-j coupling, the states are described by combinations of (j₁, j₂).

The transition from LS coupling to j-j coupling occurs as the atomic number increases. Spin-orbit coupling becomes stronger with increasing Z, making the j-j scheme more dominant.

Spectra of Many-Electron Systems

The spectra of many-electron atoms arise from electronic transitions between different energy levels. These energy levels are described by the term symbols we've discussed. The selection rules govern which transitions are allowed and thus observed in the spectrum.

Selection Rules for Electronic Transitions

For atomic spectra, the following selection rules are generally obeyed:

• ΔL = 0, ±1: The change in total orbital angular momentum must be zero or ±1. (This is for LS coupling).
• ΔS = 0: The total spin quantum number must remain unchanged. This means transitions between singlet and triplet states are forbidden (or very weak) in LS coupling.
• ΔJ = 0, ±1: The total angular momentum quantum number must change by 0 or ±1. However, a transition where ΔJ = 0 is forbidden if J=0 for both initial and final states.
• For transitions within a subshell (e.g., pⁿ), Δl = ±1.

These rules arise from the conservation of angular momentum during the absorption or emission of a photon.

Types of Spectra

The observed spectra can be emission or absorption spectra.

• Emission Spectra: Produced when excited atoms return to lower energy states, releasing photons.
• Absorption Spectra: Produced when atoms absorb photons of specific energies, causing electrons to jump to higher energy states.

Spectra of Specific Elements

The complexity of atomic spectra increases significantly with the number of electrons due to the multitude of possible energy levels and transitions.

Helium Spectrum: The 1s² configuration is the ground state, a ¹S₀ term. The first excited configuration is 1s¹2s¹, which gives rise to ¹S₀ and ³S₁ terms. Transitions from 1s¹2s¹ to 1s²: - ¹S₀ → ¹S₀: This transition is allowed (ΔL=0, ΔS=0, ΔJ=0). - ³S₁ → ¹S₀: This transition is forbidden because ΔS ≠ 0. However, due to spin-orbit coupling, this transition can occur weakly, leading to intersystem crossing. The excited configuration 1s¹2p¹ gives rise to ¹P₁ and ³P_0,1,2 terms. Transitions from 1s¹2p¹ to 1s² (¹S₀): - From ¹P₁: ΔL=1 (P→S), ΔS=0, ΔJ=1 (1→0). Allowed transitions: ¹P₁ → ¹S₀. This is responsible for the sharp ultraviolet lines in the helium spectrum. - From ³P_0,1,2: ΔL=1 (P→S), ΔS≠0 (3→1). These transitions are spin-forbidden. However, they are observed as weaker lines, especially in heavier elements. The ³P_0,1,2 terms split due to spin-orbit coupling. The helium spectrum thus exhibits two distinct series of lines: one set arising from singlet transitions (sharp, intense) and another set from triplet transitions (weaker, often broader).

Alkali Metal Spectra (e.g., Sodium): Alkali metals have one valence electron outside a closed shell (e.g., Na: [Ne] 3s¹). Their spectra are relatively simple and resemble hydrogen spectra, but with modifications due to shielding and spin-orbit coupling. The ground state is ²S_1/2. Excited states involve promoting the valence electron to higher orbitals (np, nd, etc.). For example, ns¹ → np¹. The transitions observed are primarily ²P_J → ²S_1/2. For a p electron (l=1, s=1/2), the term symbols are ²P_1/2 and ²P_3/2 due to spin-orbit splitting. The main spectral lines are: - ²P_1/2 → ²S_1/2 (ΔJ = -1) - ²P_3/2 → ²S_1/2 (ΔJ = -1) These give rise to the famous sodium doublet at 589.0 nm and 589.6 nm (the Sodium D lines).

Transition Metal Spectra: Transition metals have partially filled d or f subshells. This leads to a very large number of possible electronic configurations and term symbols, resulting in extremely complex spectra with many lines. Electron-electron interactions and spin-orbit coupling are significant.