# Equilibrium.m

Returns the thermal equilibrium state at the current temperature. Syntax:

rho=equilibrium(spin_system,H,Q,euler_angles);

Arguments:

H - Isotropic part of the Hamiltonian left side pro- duct superoperator (in Lioville space) or Hamil- tonian (in Hilbert space).

Q - 25 irreducible components of the anisotropic part of the Hamiltonian left side product superopera- tor (in Lioville space) or Hamiltonian (in Hil- bert space), as returned by hamiltonian.m

euler_angles - a row vector of Euler angles (in radians) speci- fying the system orientation relative to the in- put orientation. If the angles are not supplied, only isotropic part of the Hamiltonian is used.

WARNING: Liouville space calculations must supply left side product su- peroperators, not commutation superoperators.

WARNING: assumptions supplied to the hamiltonian.m call that generates H and Q must be 'labframe'.

If the Euler angles are not provided, uses the isotropic Hamiltonian, otherwise uses the full Hamiltonian at the specified orientation. If the temperature is set to zero during the call to create.m, returns the high-temperature approximation to the thermal equilibrium state. If the temperature is non-zero, returns the accurate equilibrium state at that temperature. This function is useful as a starting point for most NMR and EPR calculations, at both the ambient and low temperatures.

Because longitudinal spin states in Liouville space correspond to polarizations rather than populations, the state vector returned by equilibrium.m would in some cases contain small numbers (e.g. for 15N at room temperature). It is therefore advisable, when running with accurate thermal equilibria at high temperatures, to inspect the trajectory-level state space reduction tolerances and make sure that important states are not dropped automatically because of their low occupancies. The default tolerances are in most cases tight enough.

It should be stressed again that setting inter.temperature variable to be identically equal to zero (or skipping that parameter altogether) does not collapse the system into the lowest possible collective energy level, but causes equilibrium.m to return the simplified equilibrium state that is often used in basic NMR and ESR simulations, in which the equilibrium density matrix is set equal to the Hamiltonian.

Note that consistent equilibrium state normalization is only possible at finite temperatures – the high-temperature approximation makes no guarantees that the norms would be consistent between formalisms.