Chapter 6. Quantum Mechanics Part 3
Recommended Article: 【Chemistry】 Chemistry Table of Contents
1. Orbital Atomic Model
⑴ Orbital (orbital): Atomic orbital, molecular orbital, etc.
① Broad sense: Probability distribution function of electrons such as atoms, molecules in space. Also known as wave function.
② Narrow sense: The boundary surface depicted in the figure is the boundary surface when the existence probability becomes 90 %.
③ Probability density is represented by the square of the wave function ψ2.
④ The wave function is obtained from the Schrödinger equation: The reason orbitals have specific shapes is to satisfy the equation.
⑵ Quantum numbers (quantum number): Indicates the state of quantized orbitals.
① Principal quantum number: Represented by n = 1, 2, ···. Represents orbital size, energy level of electrons, number of electron shells, etc.
○ K (electron shell with n = 1), L (electron shell with n = 2), M (electron shell with n = 3), ···
② Azimuthal quantum number (Angular momentum quantum number): Represented by ℓ = 0(s), 1(p), 2(d), 3(f), ···, up to n-1. Represents the three-dimensional shape of the orbital.
○ Also represents the energy states of non-degenerated electrons.
③ Magnetic quantum number: Represented by mℓ = -ℓ, ···, 0, ···, ℓ. Represents the directional orientation of the orbital around the nucleus.
④ Spin quantum number: Represented by ms = ±1/2. Represents the direction of the electron’s spin angular momentum, i.e., the spin direction.
○ Concepts such as the phase of an orbital or the direction of electron motion within an orbital.
⑤ There are a total of 2n2 quantum states for the electron shell corresponding to n.
⑶ Nodes: The parts where the probability of electron presence becomes 0.
① Radial node: The surface at a constant distance from the nucleus where the probability of electron presence becomes 0, regardless of the direction.
② Angular momentum node: The surface where the probability of electron presence becomes 0 along a specific direction, regardless of distance.
○ The shape and number of angular nodes differ based on the type of orbital.
③ Number of Nodes
○ Radial node: n - ℓ __ - 1
○ Angular momentum node: ℓ. s orbitals have 0 nodes, p orbitals have 1 node, d orbitals have 2 nodes.
○ Total nodes: n - 1
④ The relationship between the wave function ψ(r) and the probability density function P(r)
○ P(r) = ψ(r) 2 = ψ(r)ψ*(r)
⑤ The relationship between the wave function ψ(r) and the radial probability distribution function f(r)
⑷ Shape of Atomic Orbital (Atomic Orbital, Wave Function)
① s orbital (ℓ = 0): Has a spherical electron distribution. No angular nodes. Has 1 orbital.
○ Origin: Sharp orbital
○ Absolute value of the wave function is maximum (≠ ∞) at the nucleus vicinity (r = 0), and the probability density function is 0.
○ Example: Solution of Schrödinger equation for 2s orbital (where a0 is the Bohr radius, r is the distance from the origin, c is a constant)
② p orbital (ℓ = 1): Resembles a dumbbell shape with 2 lobes. Has 3 orbitals of the same energy level (mℓ = -1, 0, 1).
○ Origin: Principal orbital
○ One lobe has a positive sign (+), the other has a negative sign (-).
○ The sign indicates phase and does not indicate the magnitude of the probability of finding an electron.
○ The sign of the phase is related to constructive and destructive interference concepts in molecular bonding.
○ Three p orbitals are oriented along the x, y, z axes and are denoted as px, py, pz.
○ Probability of finding an electron near the nucleus converges to 0.
③ d orbital (ℓ = 2): Has 4 lobes, and 5 orbitals of the same energy level (mℓ = -2, -1, 0, 1, 2).
○ Origin: Diffuse orbital
④ f orbital (ℓ = 3)
○ Origin: Fundamental orbital
⑤ Orbitals beyond f are named in alphabetical order: g (ℓ = 4) → h (ℓ = 5) → i (ℓ = 6), and so on.
⑸ Valence Electrons (Outermost Electrons)
① Number of electrons present in the outermost shell of an atom.
② Electrons participating in chemical reactions.
⑹ Selection rule: Restriction on electron transitions
① Change in principal quantum number: Δn can also be negative
② Change in angular quantum number: Δℓ = ±1
○ Example: Transition from 1s to 2p is allowed, but 1s to 2s is not allowed
③ Change in magnetic quantum number: Δmℓ = 0, ±1
④ Change in spin quantum number: During transitions, the electron’s spin does not change (conservation of spin)
⑤ Δj = ±1
2. Multielectron Atom
⑴ Breakdown of the Degenerate Orbital
① In hydrogen-like atoms, energy levels are determined only by the principal quantum number. However, for multielectron atoms, the azimuthal quantum number also contributes.
② Cause 1: Shielding Effect: As the number of electrons increases, the effective nuclear charge varies for each electron.
③ Cause 2: Penetration Effect: p electrons cannot approach the nucleus closely due to their orbital angular momentum → Increased energy.
⑵ Aufbau Principle: When orbitals are filled with electrons, they are filled from the lowest energy orbital.
① Example
Figure 1. Schematic of the Aufbau Principle
Figure 2. Schematic of the Aufbau Principle
○ 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f ＜ 6d ＜ 7p
② Note 1: According to the Aufbau principle, the 4s orbital is filled with electrons before the 3d orbital because it is lower in energy. However, once orbitals are formed, the 3d orbital becomes lower in energy than the 4s orbital, causing electrons to be removed from the 4s orbital.
○ Example: The ground state electron configuration of Fe is 1s22s22p63s23p63d64s2, and the ground state electron configuration of Fe2+ is 1s22s22p63s23p63d6.
③ Note 2: d5 and d10 orbitals have lower quantum mechanical energy than predicted.
○ Example: Cr: [Ar] 3d54s1, Cu: [Ar] 3d104s1
○ Cause 1: 4s and 3d orbitals have similar energies.
○ Cause 2: Electron-electron repulsion in the 4s orbital is unstable → Dividing electrons into the d orbitals provides stability.
⑶ Pauli’s Exclusion Principle
① The same set of four quantum numbers cannot be the same for multiple electrons in a single atom.
○ Four quantum numbers: Principal quantum number, azimuthal quantum number, magnetic quantum number, spin quantum number.
② Maximum of 2 electrons can occupy one orbital: If there are 2 electrons, their spins must be opposite.
⑷ Hund’s Rule: When there are multiple degenerate orbitals, stability increases with more unpaired electrons.
① Degeneracy: Equal energy levels.
② Paramagnetic: One or more unpaired electrons. Reacts with external magnetic field.
○ Reason: The circulation of electrons constitutes a current, resulting in Lorentz force under an external magnetic field.
③ Diamagnetic: No unpaired electrons. Does not react with external magnetic field.
④ Hund’s rule may not be satisfied if the atom’s ground state is not considered.
⑤ For hydrogen-like atoms, degeneracy is determined only by the principal quantum number n.
⑥ For sp3d2 orbitals and high-spin cases, a similar Hund’s rule applies.
3. Atomic Bonding Theory (VBT, Valence Bond Theory)
⑴ Overview: Lewis Structure - Valence Shell Electron Pair Repulsion (VSEPR) Theory - Valence Bond Theory (VBT) - Molecular Orbital Theory
① Definition: Applying orbitals to Lewis structures, VSEPR.
② Approximate quantum theory.
⑵ Assumptions: Different orbitals of one atom become hybridized to form hybrid orbitals for bonding.
① Supplement: Atomic orbitals within an atom can mix due to their wave-like properties.
⑶ σ Bond, π Bond
① The concepts of σ and π bonds started to emerge from atomic bonding theory.
② σ Bond: Bond with no nodal plane containing the bonding axis.
○ Examples: s-s, s-p, pz-pz
③ π Bond: Bond with a nodal plane containing the bonding axis.
○ Examples: px-px, py-py
④ Bond strength of σ and π bonds
○ σ bonds without nodal planes have strong bonding, π bonds with nodal planes have weak bonding.
○ Single bond: 1 σ bond. Bonding must bring benefits, so always σ bond.
○ Double bond: 1 σ bond, 1 π bond. Bond strength is less than twice that of a single bond.
○ Triple bond: 1 σ bond, 2 π bonds. Bond strength is less than three times that of a single bond.
○ π bonds need to be weak and close together → Multiple bonds are rarely observed from the third period onwards.
○ C, N, O easily form multiple bonds.
⑷ sp3 Hybrid Orbitals: Tetrahedral (e.g., ethane)
① Hybridization of 3 p orbitals forms tetrahedral orientation.
② Shapes of h1 ~ h4 orbitals can be verified directly using formulas.
○ h1 orbital: ½ (s + px + py + pz)
○ h2 orbital: ½ (s + px - py - pz)
○ h3 orbital: ½ (s - px + py - pz)
○ h4 orbital: ½ (s - px - py + pz)
③ Example: CH4
○ (initial) C atom’s orbitals: 2s2 2px1 2py1 2pz0 → Unsuitable with 2 bonding sites
○ (promotion) C atom’s orbitals: 2s1 2px1 2py1 2pz1 → Unsuitable with orbitals along axes
○ (hybridization) C atom’s orbitals: 2h11 2h21 2h31 2h41 → 4 bonding sites. Suitable for predicted shape
⑸ sp2 Hybrid Orbitals: Trigonal Planar (e.g., ethene)
① One is a σ bond, the other is a π bond in a double bond.
② Two p orbitals are hybridized, creating trigonal planar geometry.
⑹ sp hybrid orbitals: linear (e.g., ethyne, carbon dioxide)
① One is a σ bond, the other two are π bonds in a triple bond.
② One p orbital is hybridized, creating linear geometry.
③ Example: acetylene
Figure 3. Hybridization of acetylene
4. Molecular Orbital Theory (MOT)
⑴ Overview: Lewis structures - VSEPR theory - Valence bond theory - Molecular Orbital Theory (MOT)
① Limitations of Valence Bond Theory
○ Assumption: Atomic electrons reside on individual atoms.
○ Limitation 1: Cannot predict paramagnetism/diamagnetism of molecules: Lewis structure predicts O2 as diamagnetic, but it’s paramagnetic.
○ Oxygen is paramagnetic, and MOT predicts the paramagnetism due to unpaired electrons.
○ Limitation 2: Cannot predict bonding behavior: For example, why don’t noble gases form bonds?
② Characteristics
○ Fully quantum theory
○ All atomic electrons contribute to the whole molecule, not just individual atoms.
⑵ Simplification: LCAO-MO (linear combination of atomic orbital - molecular orbital)
① Definition: Approximation that molecular orbitals are formed through linear combinations of atomic orbitals’ overlap.
② Most well-known molecular orbital approximation method.
③ Approximation representing molecular orbitals as combinations (addition, subtraction) of atomic orbitals composing the molecule.
④ Number of Molecular Orbitals = Sum of atomic orbitals forming the molecule.
⑤ Number of Electrons of Atoms = Number of electrons filling molecular orbitals.
⑶ General Concepts of Molecular Orbitals
① Types of molecular orbitals based on contribution to bonding:
○ Type 1: Bonding Molecular Orbital (BO) - Lower energy than atomic orbitals
○ Constructed by reinforcing interference of atomic orbitals with the same phase.
○ Related to bonding interaction
○ Example: ψσ = ψA1s + ψB1s
○ Type 2: Anti-Bonding Molecular Orbital (ABO) - Higher energy than atomic orbitals
○ Constructed by canceling interference of atomic orbitals with different phases.
○ Related to repulsion during bonding
○ Contains nodes that hinder bonding
○ Example: ψσ* = ψA1s - ψB1s
○ Type 3: Non-Bonding Molecular Orbital (NBO) - Same energy level as atomic orbitals
② Types of molecular orbitals based on bonding:
○ σ orbitals (sigma bonds): No nodal plane containing the bonding axis
○ Example: 1sA + 1sB → σ1s + σ1s*
○ Example: 2sA + 2sB → σ2s + σ2s*
○ π orbitals (pi bonds): Nodal plane containing the bonding axis
○ Example: 2px + 2px → π2px + π*2px
○ Pi bonding typically involves the overlap of unhybridized p orbitals.
○ Pi bonds are always formed with sigma bonds because bonding is favorable.
○ Pi bonding is preferred over the formation of double sigma bonds due to electron repulsion.
○ If a pi bond breaks, the molecule becomes too unstable, so multiple bonds do not rotate.
○ For elements in the third period and beyond, the greater distance makes pi bonding difficult, resulting in fewer double bonds.
③ LUMO and HOMO
○ LUMO (Lowest Unoccupied Molecular Orbital): Lowest-energy unoccupied molecular orbital
○ HOMO (Highest Occupied Molecular Orbital): Highest-energy occupied molecular orbital
○ Band Gap Energy: Energy difference between LUMO and HOMO
○ Light corresponding to this energy transition is emitted in the ground state
○ LUMO has higher energy than HOMO
④ Bond Order
○ Bond Order = (Number of Bonding Electrons - Number of Anti-Bonding Electrons) / 2
○ Electrons in non-bonding orbitals don’t affect bond order
○ Higher bond order leads to shorter bond length and higher bond energy
○ Compare bond orders when comparing bond lengths and strengths
⑤ Number of Nodes in Molecular Orbitals: Tip. You can draw them directly
○ σ2s has 0 nodes, σ2s* has 1 node
○ σ2p has 2 nodes, σ2p* has 3 nodes
○ π2p has 1 node, π2p* has 2 nodes
Figure 4. Number of nodes in molecular orbitals
⑥ Ionization Energy and Electron Affinity
○ Lower orbital energy of removed electrons leads to higher ionization energy
○ Lower orbital energy of added electrons leads to higher electron affinity
⑷ Molecular Orbital Electron Configuration
① Same as atomic orbital
○ Electrons fill from lowest MO to highest
○ Pauli Exclusion Principle: Up to 2 electrons per MO
○ Hund’s Rule: Arrange electrons to maximize unpaired electrons
② Case 1: Bonding of H and 2nd Period Atoms (e.g., H atom on z-axis with F atom)
○ F’s 2px, 2py orbitals are non-participating, so non-bonding orbitals; energy level in HF remains constant
○ H’s 1s orbital and F’s 2pz orbital form σ orbitals, creating sp hybrid orbitals
○ Bonding orbital in σ orbitals is more contributed by F’s 2pz orbital and positioned lower than F’s 2pz orbital
○ Anti-bonding orbital in σ orbitals is more contributed by H’s 1s orbital and positioned higher than H’s 1s orbital
○ Each orbital can hold up to 2 electrons due to Pauli’s exclusion principle
③ Case 2: 2nd Period Diatomic Molecules: X2 Type Molecules
○ Orbital energy level decreases with increasing atomic number sum
○ sp orbital mixing
Figure 5. sp orbital mixing
○ σp orbital derived from 2p and σs orbital derived from 2s are similar
○ Electrons in similar orbitals repel each other according to Pauli’s exclusion principle
○ σp orbital rises, σs orbital lowers due to sp orbital mixing
○ σp orbital from 2p can rise above πp orbital due to sp orbital mixing
○ 2-1: Except for O2 and F2, significant sp hybridization occurs, switching σ2p and π2p
Figure 6. Molecular orbitals of 2nd period diatomic molecules except O2 and F2
○ 2-2: No significant sp hybridization in O2 and F2, so σ2p is smaller than π2p
Figure 7. Molecular orbitals of O2 and F2
④ Case 3: 2nd Period Diatomic Molecules
○ Contribution calculation for diatomic molecules
○ ψ = cAψA</sub< + cBψB
○ 1 = ∫ ψ 2 dτ = ∫ ψψ* dτ = cA2 ∫ ψA2 dτ + cAcB ∫ ψAψB* dτ + cAcB ∫ ψA*ψB dτ + cB2 ∫ ψB2 dτ = cA2 + cB2
○ cA2: Probability of bonding electrons residing in ψA
○ cB2: Probability of bonding electrons residing in ψB
○ Predicting Contribution for Diatomic Molecules: Assuming B is more electronegative
○ B’s higher electronegativity attracts more electrons, resulting in lower energy levels
○ Contribution to bonding orbitals: A < B. Electron density in bonding orbital is higher at B
○ Contribution to anti-bonding orbitals: A > B. Electron density in anti-bonding orbital is higher at A
○ When an electron’s orbital is close to a specific orbital of A or B, the electron is considered to be near that atom
○ Example 1: In the case of NO: Exhibits molecular orbitals similar to N2 rather than O2
⑤ Case 4: Molecular Orbitals for Polyatomic Molecules
○ Molecules with more than 3 atoms are called polyatomic molecules; the challenge lies in drawing MO diagrams.
○ 4-1: Molecular orbitals of H2O
Figure 8. Molecular orbitals of H2O
○ 4-2: Molecular orbitals of NH3
Figure 9. Molecular orbitals of NH3
○ 4-3: Molecular orbitals of BH3
Figure 10. Molecular orbitals of BH3
○ 4-4: Molecular orbitals of SiH4
Figure 11. Molecular orbitals of SiH4
○ 4-5: Molecular orbitals of CH2CHCHCH2
Figure 12. Molecular orbitals of diene
Input: 2019.03.02 18:22