--- title: "LCAO and stability of H2 molecule" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:lcao-h2-molecule-stable" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/lcao-h2-molecule-stable site: BCT Engineering --- # LCAO and stability of H2 molecule Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2078 Kartik, 8 marks - Video: [Linear Combination of Atomic Orbitals LCAO](https://www.youtube.com/watch?v=nq1zwrAIr4c) — Edmerls - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/lcao-h2-molecule-stable - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question What is linear combination of atomic orbitals (LCAO)? With the help of LCAO, justify that the "Formation of H₂ molecule is energetically stable". ## Short answer Linear combination of atomic orbitals (LCAO) forms molecular orbitals by adding atomic orbitals. For H₂, 1s±1s gives a low-energy bonding σ and a high-energy antibonding σ*. Both electrons fill σ, making the total energy 4.5 eV lower than two isolated H atoms, so formation is energetically stable. ## Answer **Linear Combination of Atomic Orbitals (LCAO)** is a method of finding molecular orbitals by adding or subtracting the atomic orbitals of the atoms that form the molecule: $$ \psi_{MO} = c_A\psi_A + c_B\psi_B $$ For two identical atoms $c_A = \pm c_B$, which gives two molecular orbitals, one bonding and one antibonding. #### Molecular orbitals of H₂ Each H atom has one electron in a 1s orbital, $\psi_{1s}(r_A)$ and $\psi_{1s}(r_B)$. When the atoms come close, these overlap and form $$ \psi_\sigma = \psi_{1s}(r_A) + \psi_{1s}(r_B) \qquad \text{(bonding)} $$ $$ \psi_{\sigma^*} = \psi_{1s}(r_A) - \psi_{1s}(r_B) \qquad \text{(antibonding)} $$ ```text psi_A + psi_B (sigma) psi_A - psi_B (sigma*) .-. .-. .-. / \_/ \ / \ ___/ ^ \___ _____/ \ ______ A charge B A \ / B builds up '-' between A, B node at mid-point ``` - **Bonding orbital:** the two waves add between the nuclei, so $|\psi_\sigma|^2$ is large there. This shared negative charge attracts both protons, so the energy $E_\sigma$ is **lower** than the energy of an isolated 1s electron. - **Antibonding orbital:** the waves cancel at the mid-point; there is no shared charge, the protons repel, so $E_{\sigma^*}$ is **higher** than the 1s energy. #### Filling the orbitals H₂ has two electrons. By the Pauli principle both can go into the lowest orbital $\sigma$ with opposite spins; $\sigma^*$ remains empty. ```text Energy | ____ sigma* (empty) | 1s _^_ / \ _^_ 1s | H_A \ / H_B | _^v_ sigma (2 e, opposite spins) ``` #### Energy versus separation The total energy of the molecule as a function of nuclear separation $R$: ```text E ^ |\ sigma* : always above zero -> repulsive | \__ | ''---...________ 0+-----------------------------------> R | \ ___...--- | \ _.-'' sigma : bonding | \ _-' | '-' <- minimum at R0 = 0.074 nm | depth = 4.5 eV ``` - At large $R$ the atoms do not interact, $E = 0$ (two separate H atoms). - As $R$ decreases, electrons in $\sigma$ lower the energy. - At very small $R$ the proton–proton repulsion dominates and the energy rises. - The minimum at $R_0 = 0.074$ nm is the bond length. The molecule's energy here is about **4.5 eV lower** than that of two separate atoms (bond energy). #### Conclusion Since the molecule, with both electrons in the bonding orbital, has **less total energy** than the separated atoms, and every system moves to its lowest energy state, the formation of the H₂ molecule is energetically favourable and H₂ is stable. Bond order $= \frac{1}{2}(2 - 0) = 1$ (a single covalent bond). Energy of 4.5 eV must be supplied to break it. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. LCAO builds a molecular orbital by: A. Multiplying atomic orbitals B. Adding or subtracting atomic orbitals of the atoms C. Averaging the nuclear positions D. Removing electrons from atoms Answer: B. psi_MO = c_A psi_A + c_B psi_B. (Key point: LCAO) 2. Why is the bonding orbital lower in energy than the 1s level? A. Nuclei move apart B. Electron charge builds up between the nuclei and attracts both protons C. The waves cancel at the mid-point D. It has fewer electrons Answer: B. Shared charge between the nuclei attracts them and lowers energy. (Key point: Bonding orbital) 3. How are the two electrons of H2 placed? A. One in sigma, one in sigma* B. Both in sigma* with parallel spins C. Both in sigma with opposite spins D. Both in sigma with parallel spins Answer: C. Pauli principle: two electrons with opposite spins occupy sigma; sigma* is empty. (Key point: Filling in H2) 4. What shows that H2 is energetically stable? A. Bond order is zero B. The energy at R0 = 0.074 nm is about 4.5 eV lower than two separate atoms C. Energy increases monotonically as R decreases D. sigma* is occupied Answer: B. Lower total energy than separated atoms means the molecule is favoured; bond order is 1. (Key point: Stability) ## Related in Electrical Engineering Material - [H2 molecule by molecular orbital theory](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ah2-molecule-mo-theory) — Electrical Engineering Material, chapter 1, asked 3× - [Bonding orbitals and instability of H3](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ah3-molecule-not-stable) — Electrical Engineering Material, chapter 1, asked 2× - [Two helium atoms: bonding and antibonding orbitals](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ahelium-atoms-bonding-orbitals) — Electrical Engineering Material, chapter 1 ## Related in other subjects - [Linear/TI/stable/memoryless: y(t), y[k]](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acheck-1-minus-e-4t-and-sin) — Data Communication, chapter 2 - [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2 - [Stable channel: Q 20, particle 4 cm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-channel-q20-particles-4cm) — Irrigation and Drainage Engineering, chapter 4 - [T/F: grid connected system more stable](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Atf-grid-connected-more-stable) — Transmission and Distribution System Design, chapter 1