Basis Choice and Representation

A useful computational basis resolves the states and observables of interest with affordable storage and controlled error. Equivalent complete representations can become very different approximations after truncation. The relevant question is therefore how accurately a particular finite subspace represents the physical calculation, not which basis is universally best.

Required background. Change of basis supplies the exact coordinate transformation; array representations distinguishes coefficients from weighted grid samples.

Finite subspaces and projection error

Let $P_N$ project onto $N$ orthonormal trial functions and let $\|\psi\|=1$. The normalized projection is $\psi_N=P_N\psi/\|P_N\psi\|$, provided the denominator is nonzero. Orthogonality splits the norm into retained and omitted weights, $1=\|P_N\psi\|^2+\|(I-P_N)\psi\|^2$. Consequently,

$$|\langle\psi\vert\psi_N\rangle|^2 =1-\|(I-P_N)\psi\|^2.$$

Thus omitted weight has a direct state interpretation. A small energy error need not imply equally small error in a sensitive observable or in an eigenvector near a degeneracy. Check the quantity actually needed and use basis-set convergence to vary the truncation.

A harmonic-oscillator basis resolves Gaussian bound states economically but can require many functions for a broad scattering state. Plane waves treat translations and kinetic energy efficiently but need high momentum cutoffs for cusps. Local finite elements adapt to spatial structure but introduce an overlap matrix. Site and occupation bases expose local interactions and symmetries. These choices change both approximation error and operator cost.

Nonorthogonal bases and the overlap matrix

For linearly independent trial functions $\phi_i$, expand $\psi=\sum_i c_i\phi_i$ and form

$$S_{ij}=\langle\phi_i\vert\phi_j\rangle, \qquad H_{ij}=\langle\phi_i\vert\hat H\vert\phi_j\rangle.$$

The denominator $c^\dagger Sc$ is the physical norm squared: two overlapping basis functions contribute cross terms. Varying the energy quotient $c^\dagger Hc/(c^\dagger Sc)$ with respect to $c^*$ gives the generalized Hermitian eigenproblem

$$Hc=ESc,\qquad c^\dagger Sc=1.$$

Here $H$ contains matrix elements, while $S^{-1}H$ represents the coefficient-space action. Confusing the two loses the physical metric. Explicitly forming $S^{-1}H$ also conceals Hermitian structure and usually worsens numerical behavior.

When $S$ is positive definite, factor $S=LL^\dagger$ and use triangular solves to construct

$$\widetilde H=L^{-1}HL^{-\dagger}, \qquad y=L^\dagger c.$$

Then $\widetilde H y=Ey$ is an ordinary Hermitian problem. The residual bound uses these orthonormal coordinates and the inverse overlap metric. It controls the finite projected problem; a residual component outside the trial space requires separate assessment. In a large sparse calculation, use a solver that handles the generalized problem without forming dense transformations.

Near dependence: a two-function example

Take normalized functions with real overlap $s$, so $S=\begin{pmatrix}1&s\\s&1\end{pmatrix}$. For $0\le s<1$, its eigenvalues are $1+s$ and $1-s$, giving

$$\kappa_2(S)=\frac{1+s}{1-s}.$$

At $s=0.999999$, the condition number is approximately $2\times10^6$. The difference of the two functions is a legitimate small-norm direction, but representing a normalized state along it requires large coefficients. Small matrix-element errors can then matter greatly. Extra basis functions have increased the nominal variational freedom while degrading arithmetic reliability.

One remedy is canonical orthogonalization: diagonalize $S=U\operatorname{diag}(s_i)U^\dagger$, retain eigenvalues above a declared threshold, and use columns $U_i/\sqrt{s_i}$. This removes directions and therefore changes the approximation space. Report the threshold, repeat the calculation over a range of thresholds, and track the target observables. A threshold justified only by a successful factorization is not an accuracy argument.

Choosing and testing a representation

Begin with the physical length, momentum, energy, symmetry, and boundary scales. Choose a family whose resolution parameter can be increased systematically. A basis that contains the initial state exactly may still represent the evolved state poorly; a localized packet can spread or develop high-frequency structure.

Compare representations on at least one shared small problem. Transform a state and observable consistently and compare expectation values before introducing truncation. Once truncation is present, compare energies, projectors, and observables rather than demanding equality of basis coefficients. Eigenvectors carry arbitrary phase; degenerate eigenspaces also permit arbitrary unitary rotations.

For unbounded operators, small $L^2$ projection error alone does not guarantee accurate energy or derivatives. Trial functions must satisfy the appropriate boundary/form-domain conditions. The Rayleigh–Ritz method states the variational setting; boundary conditions addresses its numerical implementation.

Exercises

Let $P_N\psi$ have norm $0.99$ for a normalized target. What is the fidelity of its normalized projection, and why is it not $0.99$?

Solution

The fidelity is the squared overlap, $0.99^2=0.9801$. The missing probability weight is $0.0199$. Norm and probability are different quantities.

For the two-function overlap matrix, what happens at $s=1$? Can an increased arithmetic precision repair the problem?

Solution

One eigenvalue is zero: the two functions are identical in the Hilbert-space norm. The coefficient representation is redundant. More precision cannot make an exactly singular overlap matrix positive definite; remove the redundant direction.

References

• Anderson, E., et al., LAPACK Users’ Guide, 3rd ed., SIAM, 1999, Generalized Symmetric Definite Eigenproblems — structure-preserving reduction of generalized eigenproblems.
• Golub, G. H., and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013 — conditioning, orthogonalization, and generalized spectral problems.
• Higham, N. J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002, publisher edition — error amplification by ill-conditioned representations.