Chapter 2 Outline of density matrix theory 2.1 The concept of a density matrix Aphysical exp eriment on a molecule usually consists in observing in a macroscopic 1 ensemble a quantity that dep ends on the particle's degrees of freedom (like the measure of a sp ectrum, that derives for example from the absorption of photons due to vibrational transitions). When the interactions of each molecule with any 2 other physical system are comparatively small , it can b e describ ed by a state vector j (t)i whose time evolution follows the time{dep endent Schrodinger equation i _ ^ j (t)i = iH j (t)i; (2.1) i i ^ where H is the system Hamiltonian, dep ending only on the degrees of freedom of the molecule itself. Even if negligible in rst approximation, the interactions with the environment usually bring the particles to a distribution of states where many of them b ehave in a similar way b ecause of the p erio dic or uniform prop erties of the reservoir (that can just consist of the other molecules of the gas). For numerical purp oses, this can 1 There are interesting exceptions to this, the \single molecule sp ectroscopies" [44,45]. 2 Of course with the exception of the electromagnetic or any other eld probing the system! 10 Outline of density matrix theory b e describ ed as a discrete distribution. The interaction b eing small compared to the internal dynamics of the system, the macroscopic e ect will just be the incoherent sum of the observables computed for every single group of similar particles times its 3 weight : 1 X ^ ^ < A>(t)= h (t)jAj (t)i (2.2) i i i i=0 If we skip the time dep endence, we can think of (2.2) as a thermal distribution, in this case the would b e the Boltzmann factors: i E i k T b e := (2.3) i Q where E is the energy of state i, and k and T are the Boltzmann constant and the i b temp erature of the reservoir, resp ectively . Q is the partition function de ned as 1 X E i k T b Q := e : i=0 With this formalism, the weights must b e left time indep endent b ecause we do not i have an equation for the distribution but only a formula to derive the exp ectation values (2.2). What we need is a quantum mechanical equivalent of the phase space distribu- tion or micro-canonical ensemble of classical statistical mechanics. Because of the sup erp osition principle of quantum mechanics, a simple sum generates interferences, that should not exist b etween the elements of a statistical ensemble (using again the time indep endent formalism for states in equilibrium): 1 1 X X ^ ^ ^ Aj i: a h j j i = a j i ) < A> = h jAj i = a i i j stat i i stat stat j i=0 i;j =0 To derive the quantum mechanical equivalent of a Liouville distribution, we may ^ ^ intro duce an identity op erator I on the left of A in (2.2) 1 1 1 X X X ^ ^ ^ h j ih jAj i j ih jAj i = h j < A> = i i j j i j j i i i i;j =0 j =0 i=0 3 The numb er of states is set to in nity, for a nite numb er of states N , it is enough to set the co ecients with index larger than N to zero. 2.1 The concept of a density matrix 11 and rearrange the order of the terms 1 1 X X ^ ^ < A>= h jA j ih j j i : j i i i j j=0 i=0 ^ Accordingly, exp ectation values of quantum mechanical op erators A for ensembles can be derived from the trace op eration ^ ^ hAi =tr A^ ; (2.4) where the equilibrium statistical op erator ^ is de ned as 1 X ^ = j ih j : (2.5) i i i i=0 It is p ossible to generalize the de nition (2.5) by intro ducing a basis and time de- 4 p endent co ecients : 1 X ^(t):= (t)j ih j : (2.6) ij i j i;j =0 Tohave the prop erties equivalent to a micro canonical ensemble, the density op erator 5 must b e Hermitean and p ositive semide nite b ecause when brought back to diago- nal form the co ecients (t)= (t) have the meaning of probabilities, so they ij ij i must be real and p ositive. Once the density op erator is generalized, the problem arises whether (2.6) can always b e rewritten as (2.5); b eing Hermitean, it can always b e diagonalized; but if there are states with the same p opulation, the transformation is not unique anymore. Any orthogonal set of state vectors spanning the subspace of the degenerate eigenvectors can b e used in (2.5); but all of them will give rise to the same exp ectation values as can b e veri ed by rewriting the equations for ^ and using di erent sets for the functions b elonging to the degenerate eigenvalues. So the orthogonalization pro cedure may b e problematic, but the resulting density op erator will have anyway the correct physical prop erties. If the coupling with the bath is negligible or its co ordinates are included in (2.6), the Schrodinger dynamics can b e mapp ed directly onto the density matrix space to 4 The basis set b eing complete a time dep endence on the functions is not necessary. 5 Positive semide nite op erators have eigenvalues real p ositiveornull. 12 Outline of density matrix theory get the nondissipative Liouville{von Neumann equation [37, 46] h i _ ^ ^(t)= L^(t) L ^(t)i H; ^(t) ; (2.7) H where L is the Hamiltonian Liouvillian sup erop erator. The dynamics are unitary H and the eigenvalues consequently do not change in time, keeping the distribution constant. The equations for isolated systems follow always the Hamiltonian dy- namics, b eing by de nition indep endent of external forces. Eqn.(2.7) is completely equivalent to a set of time dep endentSchrodinger equations plus the averaging (2.2). The practical solution of (2.7), however, requires the propagation of matrices rather than state vectors. This disadvantage of density matrices is outweighed only in rare cases by the advantage that b oth the propagation of mixed and pure initial states can be done with the same e ort. When the strength of the interactions with the external mo des cannot be ne- glected (like in condensed phases) each molecule will b ehave according to the slightly di erent environment it is in, thus mo difying the distribution (2.6). The dynamics is even more p erturb ed when we lo ok at the prop erties of small parts of molecules like the chromophores of a macromolecule or the absorption prop erties of the ac- 6 tive groups of chlorophyll or rho dopsin, b oth emb edded in large biological systems. Here with the time evolution app ears the phenomenon of IVR (Intramolecular Vi- brational Redistribution). This could be mo deled using a distribution of molecular Hamiltonians (source of the inhomogeneous broadening in sp ectroscopy) and/or uctuating forces (homogeneous broadening). All these phenomena can again be more elegantly, and often more eciently, be handled in the framework of density matrix theory. Here, the bath can b e left out of the dynamics, but its e ects on the system can b e repro duced directly in the equations of motion of the system via the (non-Markovian) dissipative Liouville-von Neumann equation [37] Z h i t _ ^ ^(t)=L^(t)= i H ; ^(t) + d L (t; )f^( )g: (2.8) s D 0 The relevant observables are then directly computed with ^(t) and relation (2.4). ^ In (2.8), the Hamiltonian is indicated with H to stress that it refers only to the s co ordinates of the smaller system. One usually talks ab out the \reduced dynamics" 6 Molecules involved in the photosynthesis and in the visual pro cess resp ectively. 2.2 Equations of motion 13 ^ of the \reduced density op erator". H is an e ective Hamiltonian b ecause even if s it dep ends only on the system mo des, it includes the static (averaged) distortion of the system dynamics that is due to the environmental degrees of freedom, like the additional electrostatic eld of a crystal cavity. The term Z t d L (t; )f^( )g D 0 enters in (2.8) as a dissipative corrective to the Hamiltonian evolution to describ e the dynamical coupling of the system to an unobserved environment and dep ends on the uctuations of the bath variables. It accounts for energy and phase relaxation,i.e., the mo di cation of distribution (2.5), or equivalently, of the eigenvalues of (2.6). The derivation of the dissipative term may lead to a non-Markovian evolution (as in Eqn.(2.8)), then the time evolution of the density op erator dep ends up on its past, the system develops a \memory". Under the Markov approximation (i.e., when memory e ects are neglected), the equations of motion turn into [37, 46, 4]: h i _ ^ ^(t)=L^(t)=i H ; ^(t) +L (^(t)) : (2.9) s D Here, L (^(t)) is a function of ^ at time t only and is usually linear.