Every problem, one list

Pick a parameterized trial wavefunction, write the energy as a function of the parameters, and minimize — the upper-bound calculation at the root of every electronic-structure method.

1. 1workedHelium in one parameter — every integral shown
2. 2practiceWhen the trial contains the exact answer
3. 3practiceA basis of the wrong shape
4. 4practiceA parameter-free trial on a different system
5. 5practiceReading a physical constant out of the optimum
6. 6practiceLinear variation — the secular equation
7. 7checkAn energy below exact
8. 8checkWhy the bound is for the ground state only

Evaluate molecular integrals over s-type Gaussians by hand via the Gaussian product theorem.

1. 9practicegeometry / weighted averages
2. 10practicescreening & sparsity
3. 11practicebasis design
4. 12practiceoperator limits
5. 13practicenormalization
6. 14practicespecial functions
7. 15checkCheck 1
8. 16checkCheck 2

Constrained functional variation: minimize the orbital energy functional under orthonormality and earn the Fock eigenvalue equation.

1. 17workedFrom the energy functional to the eigenvalue equation
2. 18practiceCalculus of variations — where the 2 comes from
3. 19practiceVarying the Coulomb functional
4. 20practiceBrillouin as a unit test
5. 21practiceThe unitary freedom
6. 22practiceWhy “density stopped” equals “energy stationary”
7. 23practiceKoopmans from the canonical equations
8. 24checkThe FPS − SPF criterion
9. 25checkMultipliers vs. energies

Recognize a problem as a fixed-point iteration — an operator that depends on its own eigenvector — and solve by iterative refinement: linearize with a guess, solve, use the result to rebuild the operator, iterate. Diagnose convergence, oscillation, and the use of damping or DIIS-style acceleration.

1. 26practicescalar fixed-point analysis
2. 27practicecosine fixed-point (Dottie number)
3. 28practicepower iteration for eigenvectors
4. 29practiceoscillation and damping
5. 30practicemean-field self-consistency (Ising)
6. 31practiceSCF convergence diagnostics
7. 32practicecross-domain self-consistency
8. 33checkCheck 1
9. 34checkCheck 2

Reduce determinant energies to one- and two-electron integrals; let spin orthogonality kill terms.

1. 35practicereal H₂ numbers
2. 36practiceorbital energies
3. 37practicetriplet spin
4. 38practicefour electrons
5. 39practiceKoopmans by subtraction
6. 40practiceterm bookkeeping
7. 41checkCheck 1
8. 42checkCheck 2

Evaluate Hamiltonian matrix elements between determinants by counting orbital differences — three rules replace N! terms.

1. 43practicediagonal energies
2. 44practiceBrillouin numerically
3. 45practicethe MP2/FCI coupling
4. 46practicethe two-body cutoff
5. 47practicepermutation signs
6. 48practicespin selection rules
7. 49checkCheck 1
8. 50checkCheck 2

Evaluate matrix elements with creation/annihilation anticommutators; normal ordering and Wick contractions.

1. 51practicenumber operator
2. 52practicePauli algebraically
3. 53practiceone-body element
4. 54practiceordering signs
5. 55practicenormal order
6. 56practiceWick contractions
7. 57checkCheck 1
8. 58checkCheck 2

Normalize Slater and Gaussian primitives, read the exponent as an inverse size, and use the Gaussian product theorem — the bookkeeping behind every basis set.

1. 59workedNormalize a primitive 1s Gaussian
2. 60practiceThe Slater orbital, for contrast
3. 61practiceThe cusp a Gaussian can never make
4. 62practiceExponent as inverse size
5. 63practiceThe product theorem, with numbers
6. 64practiceCounting the degrees of freedom
7. 65checkDerive the Gaussian product theorem
8. 66checkWhy Gaussians win anyway