From Quantum Mechanics to Materials

A material model is a simplified description of interacting electrons and nuclei, chosen for a particular question. Bands, phonons, local moments, and screened interactions emerge through different approximations. Understanding those steps tells you both why a model works and what changes when you ask it to predict a different observable.

This page follows the reduction from microscopic quantum mechanics to a working material model, using low-field transport in silicon as an example. Choosing a Model compares possible starting descriptions; Computational Quantum Matter explains how to solve and test a chosen one.

Helpful background. The time-independent Schrödinger equation supplies the eigenvalue language, while identical particles explains fermionic antisymmetry. The detailed reduction methods are developed in effective Hamiltonians and scale separation.

Electrons and nuclei are the starting point

A nonrelativistic Coulomb model contains

$$H=T_{\mathrm e}+T_{\mathrm I} +V_{\mathrm{eI}}+V_{\mathrm{ee}}+V_{\mathrm{II}}.$$

The two kinetic energies describe electrons and ions; the three potentials describe electron–ion, electron–electron, and ion–ion interactions. External fields and relevant relativistic corrections, such as spin–orbit coupling, can be added. If an environment matters, either include its degrees of freedom or specify the approximation used to eliminate them.

The electronic wavefunction is antisymmetric under exchange. That restricts allowed states but does not replace the Coulomb interaction: exchange statistics and dynamical correlation are distinct. A Slater determinant respects antisymmetry while generally approximating the interacting state.

Likewise, translation symmetry can give an interacting many-electron state a total crystal momentum without making it a product of independent Bloch orbitals. A useful one-electron equation requires an additional approximation or effective description.

Separate electronic and nuclear motion

At fixed nuclear positions $\mathbf R$, solve the conditional electronic problem

$$H_{\mathrm e}(\mathbf R)\Phi_\alpha =E_\alpha(\mathbf R)\Phi_\alpha.$$

Here $H_{\mathrm e}$ includes electronic kinetic energy and electron–ion and electron–electron interactions. The surface governing nuclear motion also includes ion–ion repulsion:

$$U_\alpha(\mathbf R)=E_\alpha(\mathbf R)+V_{\mathrm{II}}(\mathbf R).$$

Although $V_{\mathrm{II}}$ is constant during the electronic calculation, it changes with nuclear positions and contributes to forces. Omitting it gives the wrong equilibrium geometry.

The Born–Oppenheimer approximation exploits electronic–nuclear scale separation to neglect or systematically treat couplings between electronic surfaces. Nuclear motion can remain fully quantum on a retained surface. Small electronic gaps, strong derivative couplings, or fast driving can invalidate the separation. Replacing a real finite sample by an infinite perfect crystal is a further assumption, separate from Born–Oppenheimer theory.

Understand what a calculated band means

Hartree–Fock, Kohn–Sham, fitted tight-binding, and quasiparticle methods can all produce eigenvalue plots resembling bands. Their eigenvalues have different meanings.

Kohn–Sham density-functional theory uses an auxiliary noninteracting system to reproduce the ground-state density. Its individual eigenvalues are not generically exact charged-excitation energies. A quasiparticle treatment instead seeks poles of an interacting propagator and can involve an energy-dependent, nonlocal operator. Self-consistency alone does not establish that either approximation captures every correlation effect.

For a periodic one-body operator, Bloch’s theorem is an exact symmetry decomposition. The construction of that operator and the physical interpretation of its eigenvalues are separate questions. Band Theory Overview explains this distinction in detail.

Restore lattice vibrations when the observable needs them

Expand an adiabatic surface near a stable equilibrium configuration:

$$U(\mathbf R^0+\mathbf u)-U(\mathbf R^0) =\frac12\sum_{I\alpha,J\beta} \Phi_{I\alpha,J\beta}u_{I\alpha}u_{J\beta}+\cdots.$$

The force constants $\Phi$ and ionic masses determine the normal modes. Diagonalizing this quadratic problem is exact for the chosen harmonic Hamiltonian; neglecting the higher powers of displacement is the approximation. Quantization of its positive-frequency modes gives phonons. Uniform translation of an unpinned finite crystal is a free center-of-mass coordinate, not a positive-frequency oscillator.

Translational invariance requires the acoustic sum rule

$$\sum_{J}\Phi_{I\alpha,J\beta}=0,$$

where $J$ includes every ion in every cell. A substrate or external pinning can change this condition. In a free-crystal calculation, a violation can indicate inconsistent force constants or numerical truncation.

Electronic matrix elements also change with displacement. If $Q_\lambda$ is a normal coordinate,

$$t_{ab}(Q)=t_{ab}^{(0)} +\sum_\lambda \left.\frac{\partial t_{ab}}{\partial Q_\lambda}\right|_0Q_\lambda+\cdots.$$

Quantizing the coordinate produces an electron–phonon coupling. Thus a fixed-ion electronic calculation and a phonon calculation are compatible stages of the same description. Electron–phonon coupling connects these derivatives to scattering and other observables.

Select a band subspace and its interactions

A unitary change from Bloch states to Wannier functions preserves information within a complete selected subspace. Smooth, exponentially localized Wannier functions require additional regularity and topology conditions; a nonzero Chern number obstructs such a basis for that subspace. Selecting only some bands, discarding distant matrix elements, or replacing a frequency-dependent interaction by a constant are further approximations. See Wannier Functions.

Screening also depends on which electronic processes remain explicit. In a constrained random-phase construction, write $P=P_{\mathcal T}+P_r$, where $P_{\mathcal T}$ contains the polarization processes to be treated by the retained low-energy model. Use the potential-energy convention $P=\delta n/\delta U_{\mathrm{tot}}$; a stable static density channel has negative polarization. The interaction screened by the remaining processes is

$$W_r(\omega)=[1-vP_r(\omega)]^{-1}v,$$

with bare Coulomb kernel $v$. Products can be matrices, so their order matters. If the low-energy solver subsequently treats $P_{\mathcal T}$, the result is

$$W=[1-W_rP_{\mathcal T}]^{-1}W_r.$$

Using the fully screened $W$ as the input to that same screening calculation would count the target polarization twice. Aryasetiawan and colleagues develop the frequency-dependent effective-interaction construction. A single static on-site $U$ is an additional simplification, whose usefulness depends on the selected orbitals and energy window.

This is one reason parameters cannot be transferred casually between models. An interaction computed for one orbital subspace may not be consistent with hoppings or a self-energy computed for another. The random-phase approximation gives the response conventions and derivation.

Example: low-field electrons in silicon

Consider a homogeneous, unstrained, bulk n-type silicon sample in a weak electric field, at temperatures and densities where conduction electrons are nondegenerate and a semiclassical transport description is appropriate. The aim is to predict conductivity, not every property of silicon.

1. Retain the relevant electronic states. Unstrained silicon has six symmetry-related conduction minima along the equivalent $\langle100\rangle$ directions. Near a minimum $\mathbf k_{0\nu}$, approximate the dispersion by

$$E_\nu(\mathbf k_{0\nu}+\mathbf q)-E_c =\frac{\hbar^2}{2}\sum_{ij}q_i(m_\nu^{*-1})_{ij}q_j+\cdots.$$

The valley label $\nu$ and anisotropic effective-mass tensor remain explicit. The masses summarize local curvature; they are not bare electron masses. The approximation becomes inadequate when occupied states explore substantial nonparabolicity, remote bands, or strong confinement. Effective Mass develops the expansion.

2. Include the processes that relax current. Phonons and ionized impurities can scatter carriers. Intervalley scattering is a process to retain when the temperature and carrier energies activate it, not automatically a failure of the multivalley description. A relaxation time can depend on energy, temperature, valley, and impurity density; it is not a universal material constant.

As a simple illustrative limit, give all six equally populated valleys the same scalar relaxation time $\tau$. Their longitudinal and transverse masses are $m_l$ and $m_t$. Summing the rotated inverse-mass tensors gives an isotropic conductivity,

$$\sigma=\frac{ne^2\tau}{m_c},\qquad \frac1{m_c}=\frac13\left(\frac1{m_l}+\frac2{m_t}\right).$$

The conductivity mass $m_c$ is not the density-of-states mass. Unequal valley populations, strain, or anisotropic collision processes require the more general tensor calculation.

3. Check the transport regime. Nondegeneracy requires carrier density well below the effective conduction-band density of states. Semiclassical motion also needs well-defined carrier states and scattering that does not destroy the band description; a directional thermal-wave-vector criterion is $k_{\mathrm{th}}\ell\gg1$. For bulk diffusive transport, the relevant mean free paths must be small compared with sample dimensions. The field must be weak enough to avoid appreciable heating or a strongly distorted distribution. Boltzmann Transport explains the collision equation beyond the single-$\tau$ estimate.

4. Relate conductivity to the measured voltage. Contact arrangement, current path, sample dimensions, and thermometry enter the resistance measurement. A converged band calculation does not test these ingredients. Compare predicted temperature and density dependences with observations not already used to fit the scattering parameters.

If the question changes to absorption near silicon’s indirect optical edge, the retained ingredients change too: photon and phonon matrix elements, electron–hole attraction, selection rules, and optical resolution become necessary. A lifetime fitted to dc mobility cannot simply be reused for that optical process.

Keep the observable consistent with the approximation

Eliminating states changes more than the Hamiltonian. It can also change the effective current, density, spin, or dipole operator. An accurate low-energy spectrum does not automatically imply accurate transition strengths.

For example, projecting out charge fluctuations can produce a spin Hamiltonian for a half-filled Mott insulator. The leading single-band exchange is $J=4|t|^2/U$ when $U\gg|t|$ and the other assumptions of the reduction hold. This model describes low-energy spin dynamics but cannot represent optical excitation across the eliminated charge gap. The effective-Hamiltonian derivation explains how observables are transformed along with states.

When assessing a result, distinguish three questions: is the physical model appropriate, is the retained basis or system size adequate, and has the resulting numerical problem been solved accurately? Improving the last answer cannot repair the first two. A useful model comparison tests an observable, a symmetry or conservation law, and a limit with a known answer.

Exercises

1. Fixed nuclei and forces. Why is $V_{\mathrm{II}}$ irrelevant to electronic eigenvectors at fixed nuclear coordinates but essential when locating equilibrium positions?

Solution

At fixed $\mathbf R$, the ion–ion energy is a scalar in the electronic Hilbert space, so adding it shifts every electronic eigenvalue equally. Its derivative with respect to $\mathbf R$ contributes to nuclear forces. The equilibrium condition involves derivatives of $E_\alpha(\mathbf R)+V_{\mathrm{II}}(\mathbf R)$, together with any other retained coordinate-dependent terms.

2. Translation without independence. Why does a translation-invariant electron interaction conserve total crystal momentum without making every electron momentum separately conserved?

Solution

Simultaneous lattice translation of all electrons leaves their relative coordinates unchanged. The Hamiltonian therefore commutes with total lattice translation. An interaction can nevertheless scatter a pair between individual momenta, conserving their sum modulo a reciprocal vector. Total-momentum labeling does not imply a product state or independent motion.

3. Screening in two stages. In a scalar static example take $v=4$, $P_r=-0.10$, and $P_{\mathcal T}=-0.15$ in compatible units. Calculate $W_r$ and $W$. What happens if the target polarization is applied a second time?

Solution

$W_r=4/(1+0.4)=20/7$. Including the target processes gives $W=(20/7)/(1+3/7)=2$, the same as $4/[1-4(-0.25)]$. Screening that result again by $P_{\mathcal T}$ gives $2/(1+0.3)=20/13$, which is too small for the stated model because the target processes were counted twice.

4. Six anisotropic valleys. Explain why equal populations and a common scalar $\tau$ produce the isotropic conductivity mass above.

Solution

Two valleys have their longitudinal axis along each Cartesian direction. For any chosen direction, two of the six inverse masses are $1/m_l$ and four are $1/m_t$. Each valley carries density $n/6$, so the summed coefficient is $ne^2\tau[(1/m_l+2/m_t)/3]$. Cubic symmetry makes the three diagonal components equal; off-diagonal components vanish in this principal-axis model.

5. Low energy is not enough. A target energy is much smaller than the separation to an omitted band, but an interaction matrix element coupling the two bands is comparable to that separation. Is simply discarding the second band justified?

Solution

No. The coupling can mix the bands strongly even when the desired excitation energy is small. Compare coupling matrix elements with the separation, use a controlled elimination where possible, or check a larger retained space. An effective Hamiltonian and its observables can acquire substantial corrections from the omitted band.

References

• F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein, “Frequency-dependent local interactions and low-energy effective models from electronic structure calculations,” Physical Review B 70, 195104 (2004), doi:10.1103/PhysRevB.70.195104.
• M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, 1954.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015, doi:10.1017/CBO9781139020916.
• W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Physical Review 140, A1133–A1138 (1965), doi:10.1103/PhysRev.140.A1133.
• P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Materials Properties, 4th ed., Springer, 2010, doi:10.1007/978-3-642-00710-1.