Quantum theory of molecular manyparticle systems
 Post by: Bjoern Baumeier
 December 24, 2018
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I. Quantum theory of molecular manyparticle systems
These notes are part of a full lecture about “Charge and Energy Transfer Dynamics in Molecular Systems“. See its full Table of Contents.
Table of Contents
 Molecular Schrödinger Equation
 BornOppenheimer Separation and the adiabatic approximation
 Electronic structure methods: overview
 Noninteracting singleparticles
 The HartreeFock method
 DensityFunctional Theory
 Comparison of singleparticle and excitation energies
 Fermi’s Golden Rule
 Harmonic approximation for nuclear motion
I.1 Molecular Schrödinger Equation
For the moment, let us consider a rather general picture of a molecule as indicated in Figure ??: an aggregate structure consisting of nuclei and electrons. The coordinates (spins of the individual nuclei and of the individual electrons are combined into the supervariables and respectively.
The full quantummechanical information about the molecule is — at least in principle — obtained by solving the timedependent Schrödinger equation
(1)
The Hamiltonian of the molecule is (in the absence of external fields) given by
(2)
The Hamiltonian in Eq.2
 is except for relativistic corrections exact;
 does not explicitly depend on the time
 is exactly solvable only for (hydrogen atom)
 “contains” all phenomena, such as
 structural properties (equilibrium geometries)
 dynamical properties (vibrations)
 electronic properties (ionization spectra, UPS)
 optical properties (absorption, emission)
 transport properties
 …
From ii. it follows that wave function as the solution of the timedependent Schrödinger equation Eq.1 factorizes as
(3)
where and and are obtained as solutions of the stationary Schrödinger equation
(4)
The task that we face in describing charge and energy transfer processes is clear: solve Eq.4 and extract the relevant properties from the solution. Obviously, this is impossible to achieve exactly and instead has to rely on physically sensible approximations.
I.2 BornOppenheimer Separation and the adiabatic approximation
The first approximation to introduce deals with the relevant time scales of nuclear and electronic motion. Two rather simple arguments can be put forward to motivate this:
 We know that stable molecules exist!
 From this very basic realization, it follows that the forces acting on the electrons and the nuclei must be similar in this stable equilibrium:
(5)
As a consequence, the dynamics of the nuclei is much slower than that of the electrons. In other words, the electrons adjust instantaneously to the nuclear motion, i.e., the electrons move adiabatically.
To express this situation in formal terms, we consider a fixed arrangement of nuclei The electronic Hamiltonian representing the electronic system that interacts with the fixed nuclear configuration reads
(6)
In this situation, is no longer a variable of the electronic system, but a fixed parameter for the electronic degrees of freedom. The corresponding stationary electronic Schrödinger equation is given by
(7)
where is a set of adiabatic electronic wave functions. Those can be used as a basis to expand the molecular wave function according to
(8)
Entering this BornOppenheimer separated wave function into Eq.4 yield
(9)
If we now first multiply this equation by and then integrate over the electronic coordinates, we obtain an expression for the nuclear functions
(10)
One has to pay special attention evaluating the action of the kinetic energy operator of the nuclei on the terms of the adiabatic wave function, marked red, within the integral above. Using the product rule, one finds
(11)
We can now enter this result into the integral in Eq.10
(12)
The remaining integral
(13)
defines matrix elements of the transition between electronic states and induced by the dynamics of the nuclei. This means that in order to determine the functions one has to solve the coupled set of equations
(14)
In the adiabatic approximation it is assumed that i.e., there are no transitions between different electronic states, and the nuclear motion for each electronic state is determined by
(15)
This describes the motion of the nuclei in an effective potential
(16)
which is also called the potential energy surface (PES). Note that for each electronic states there is a unique PES associated with it, as indicated in Fig.??.
In practical calculations, one has to
 Fix nuclear coordinates and solve the electronic problem in Eq.7, obtaining and
 Solve Eq.15 with the effective potential
I.3 Electronic structure methods: an overview
In this part, we will take a look at fundamental approaches for the solution of the electronic problem commonly used in quantum chemistry. It is obvious that at this point we cannot give a fully detailed account of those methods and refer to standard textbooks (e.g., C.J. Kramer, “Essentials of Computational Chemistry”, 2nd Edition, Wiley, 2008) for an indepth discussion. The purpose of this part is instead to convey the basic ideas, introduce typically used terminology, and illustrate the strength and weaknesses for the various approaches.
What we seek are the solutions of the electron Schrödinger equation 7
(17)
where the are adiabatic wave functions of interacting electrons and the corresponding Hamiltonian is
(18)
One can clearly see that the electronic Hamiltonian comprises a sum of singleparticle Hamiltonians and a manybody Coulomb term The various approaches presented in the following essentially all differ in the approximations made to take this manybody interaction into account.
Noninteracting singleparticles
The simplest possible approximation that one can make is the limit of vanishing Coulomb interactions, or noninteracting singleparticles. In this case, the electronic Hamiltonian consists solely of the sum of singleparticle terms
(19)
and the corresponding electron wave function is simply a product of singleparticle functions
(20)
These singleparticle functions are solutions to the singleparticle Schrödinger equation
(21)
where is a singleparticle energy.
What we have now managed is to transform a single equation for (noninteracting) particles to equations of singleparticles, which are obviously much easier to treat. The total energy of the noninteracting particle system is given by
(22)
i.e., a sum of the respective singleparticle energies. Note that denotes a set of singleparticle quantumnumbers When forming the product function in Eq.20 one only has to take the Pauli exclusionprinciple into account.
One can easily form the configuration of minimal , i.e. the ground state energy by sorting all according to their value and the filling up each energy level starting from the bottom with two electrons of opposite spin, see Fig.??. The singleparticle function of the last filled energy level is called the highest occupied molecular orbital (HOMO), the one of the first empty level the lowest unoccupied molecular orbital (LUMO).
The beauty of this approach is that one can straightforwardly relate important quantities from the noninteracting singleparticle energies:

 electron removal
If we remove the electron, the energy of the electron system is(23)
The occupied singleparticle energies are therefore the negative of the respective ionization energy.
 electron addition
If we add an electron to the energy level, the energy of the electron system is(24)
The unoccupied singleparticle energies are therefore the negative of the respective electron affinities (though the definition of the sign may differ).
 electron promotion
 electron removal
If we promote an electron from an occupied level to the previously empty energy level (e.g. by optical excitation), the total energy of is
(25)
So the, e.g. energy of absorption, is equal to the difference in singleparticle energies.
Often, basic details of the electronic structure of molecules related to charge and energy transfer processes are discussed in term of this singleparticle picture. However, the downside is that, of course, the electrons of realistic molecular systems do interact. Still, one can certain within limits stay within this framework taking the interactions of the electrons into account by transforming the full interacting problem into what is called an effective singleparticle problem, as will be shown in the following.
The HartreeFock method
One aspect that we have not addressed in the previous section is the explicit Fermion nature of the electrons that requires the electron wave function to be antisymmetric with regards to particle exchange. This requirement is not fulfilled by the simple product ansatz in Eq.20. One can, however, construct an appropriate function from singleparticle functions using a determinant ansatz
(26)
often referred to as Slater determinant. Now, instead of starting from predetermined singleparticle functions and enforcing antisymmetry, we can start from the requirement of antisymmetry and use the variational principle to derive a set of equations that determine suitable effective singleparticles for the interacting case. This is the principle idea of the HartreeFock method. Let us focus on the ground state energy and consider a Slater determinant as in Eq.26. From the variational principle it holds that the energy as a functional of that determinant approximates the true ground state energy from above, i.e.,
(27)
The task is now to minimize this functional by means of a nonlinear variation with respect to the (undetermined) singleparticle functions under the condition that those functions are normalized (introduction of Langragian multipliers )
(28)
This variation can be performed using a functional derivative
(29)
where we introduced as a variable containing and and suppressed the superscript of the singleparticle functions. After some algebra (see textbooks), one arrives at a set of equations that allow to determine the
(30)
Introducing the densities
(31)
the HartreeFock equations read
(32)
While the first integral in Eq.32 corresponds to the classical Hartree integral of the Coulomb interaction
(33)
the second integral defines the exchange potential in operator form
(34)
that is a direct result of the quantummechanical antisymmetry condition of the wave function. The electron problem has thus been mapped on a set of effective singleparticle problems with the HartreeFock potential
(35)
The total energy of the ground state is, however, not simply the sum of the “singleparticle energies” (remember that these were introduced as Lagrangian multipliers during the variation) due to a double counting of interactions in and Therefore the total energy reads
(36)
where
(37)
In terms of the meaning of the , one finds

 electron removal
If we remove the electron, the energy of the electron system is(38)
The occupied singleparticle energies are therefore the negative of the respective ionization energy. This is Koopman’s theorem.
 electron addition
If we add an electron to the energy level, the energy of the electron system is(39)
The unoccupied singleparticle energies are therefore the negative of the respective electron affinities (though the definition of the sign may differ). However, since the unoccupied states do not contribute to the exchange term in the HartreeFock equation, the respective energies contain a spurious repulsive electronic selfinteraction resulting from the Hartree term. For the occupied states, this is compensated by the selfinteraction in the exchange term!
 electron promotion
 electron removal
If we promote an electron from an occupied level to the previously empty energy level (e.g. by optical excitation), one can show that the total energy is
(40)
So the, e.g. energy of absorption, can be approximated by the difference in singleparticle energies (with the limitations mentioned above).
DensityFunctional Theory
Instead of dealing with the manybody wave function directly, two theorems by Hohenberg and Kohn relate the ground state to the electron density:
 The ground state is a onetoone functional of the particle density Here, is only a single spatial coordinate!
 The energy functional
(41)
obeys a variational principle with respect to the particle density and is minimal for the ground state density :
(42)
Again, we introduce singleparticle wave functions to express the density
(43)
and proceed by separating the classical Coulomb effects (the Hartree energy) from any quantummechanical exchangecorrelation interactions of the electrons
(44)
Variation of with respect to can be rewritten as a functional derivative with respect to under the constraint of normalization
(45)
(formally identical procedure as in HartreeFock) and we again obtain a set of effective singleparticle equations, the KohnSham equations
(46)
The practical problem of applying these equations to obtain a formally exact singleparticle representation of the electron ground state lies in the fact that the exchangecorrelation functional that contains all quantum interactions is unknown, ans with it also the exchangecorrelation potential
(47)
in Eq.46. In practice, one therefore has to resort to physically motivated approximations to , such as
 Localdensity approximation (LDA)
Here one assumes that the real electron density at a certain point can be locally approximated by that of the freeelectron gas, for which is known and can be parametrized.  Generalizedgradient approximation (GGA, PBE, …)
As an extension to the LDA, the local gradient of the electron density is taken additionally into account.  hybrid functionals (B3LYP, PBE0, …)
Here, a mix of local or semilocal functionals with exact exchange contributions (see HartreeFock) is used.
Irrespective of that actual functional, we can determine the ground state energy again by removing the electronelectron interaction double counting
(48)
What is the meaning of the ? Due to the similarities between the HartreeFock and KohnSham equations one is inclined to assume that also the KohnSham eigenvalues can be directly related to excitation energies as the ionization potentials or electron affinities. However, there is no full formal justification for this, and one can only show that
 for exact DFT, meaning for the exact or the highest occupied KohnSham eigenvalue is the negative of the ionization potential
(49)
and similarly
(50)
 in general, the eigenvalue correspond to the change of the total energy with respect to a change of occupation of the KohnSham orbital (Janak’s theorem)
(51)
Comparison of singleparticle and excitation energies
 Example 1: Carbonmonoxide
Exp. HF DFTPBE DFTB3LYP IP
EA14.0
1.313.1
3.213.8
2.814.2
2.619.8 21.9 13.9 15.7 17.0 17.4 11.7 13.1 14.0 15.0 8.8 10.4 1.3 +4.0 1.7 0.8 The above table lists results of several HartreeFock or KohnSham energies for a single CO molecule in vacuum compared to experimental energies (PUT REF). One can see that the HartreeFock energies for the occupied states agree within 12 eV with the experimental reference. So, while Koopman’s theorem allows to make a formal link between the and the ionization energies, the lack of electron correlation in the effective potential leads to a disagreement with measurements. The agreement of the LUMO and higher energies is usually poor. In the DFT results with both a semilocal (PBE) and hybrid (B3LYP) functional, one finds that both (massively) underestimate the experimental data. For the occupied levels, there is an incomplete cancellation of the repulsive Coulomb selfinteraction in the Hartree term by the approximate exchangecorrelation potential, yielding too high energies. Also the too repulsive potential can lead to an overdelocalization of the singleparticle orbitals.
To obtain more reliable values for the ionization potential and electron affinity from any effective singleparticle theory, it is advised not to use the but determine the total energie differences (this is often referred to as the approach):(52)
The resulting energies are listed at the bottom of the table. Here, we see that the DFT IPs agree very well with the reference value, and the influence of the different functionals seems to be on the order of 0.1eV, roughly an order of magnitude smaller than on the .
Regarding the electron affinity, all calculations predict a negative electron affinity, i.e., they predict that CO does not want to add an additional electron, which contradicts experimental observations. In fact, the electron affinity of CO is very unique and seems to be a real problem even for higher order quantumchemical methods.  Example: Fullerene C_{60}
Exp. HF DFTPBE DFTB3LYP IP
EA7.6
2.77.4
1.17.5
2.67.6
2.59.0 9.7 6.9 7.6 7.6 7.9 5.8 6.3 2.7 0.8 4.1 3.5 For a considerably bigger molecule, such as C_{60}, we find basically the same trends as in CO with respect to the accuracy of the . DFTbased calculations, on the other hand, agree within 0.2eV with the reference data. The respective HartreeFock electron affinity in contrast is off by more than 1eV.
Lessons:
 No method is truly exact!
 HartreeFock has exact exchange but neglects electron correlations
 energies of occupied orbitals are in reasonable agreement with experimental ionization energies
 unoccupied (virtual) orbital energies are a bad estimate for EAs
 DFT is only a formally exact representation of the manybody problem, but we “cheated” by putting all quantum effects into the unknown exchangecorrelation functional. Due to the approximate nature of the approximate potentials, we introduce errors, typically:

 occupied energy levels underestimate the IPs
 unoccupied energy levels underestimate the EAs
about underestimate of HOMOLUMO gaps

 HOMO and LUMO energies can be related to electron removal and addition energies. They can be interpreted as representative levels for hole/electron transfer, though a more reliable estimate is obtained by a calculation.
 The HOMOLUMO gap (or IPEA) is not the excitation energy relevant for, e.g., photoabsorption. Such an absorption process is the promotion of one electron from the HOMO to the LUMO (in the most simple case). This is not identical to an independent removal of an electron from the HOMO and addition to the LUMO.Example: C_{60} from experiment
(53)
The difference between the free interlevel transition (the HOMOLUMO gap) and the excitation energy is the electronhole binding energy. An optical excitation, i.e., a coupled twoparticle excitation, can never be described by independent effective singleparticles.
 Apart from overinterpreting the quantitative results of the one has to be careful about
 PES for vanderWaals bonded systems, e.g., a benzene dimer. Here, HF for instance does not predict any bound dimer configuration (as a function of separation). DFTPBE does in this case but in general is not reliable;
 molecules with strongly structures electron densities is larger molecular structures, such as donoracceptor architectures or molecules containing long conjugated systems, due to DFT overdelocalization;
 not introducing new physics/chemistry by using inappropriate computational parameters, like choice of the DFT functional, basis sets, point samplings, convergence criteria etc.
I.4 Fermi’s Golden Rule
In the following, we want to consider a twolevel system. The two states and shall correspond to two eigenstates of the system, in which subsystems 1 and 2 do not interact (diabatic states). Assuming that the system is initially prepared in state , we would like to know the rate of transfer into state if we now switch on a (small) interaction.
First, we can express the Hamiltonian of the full system in the basis of the two diabatic states
(54)
where and are the energies of the two independent subsystems and is the coupling element between them. The fact that the system is supposed to be initially in state can be described in terms of the occupation probability . Evidently, . The question now is how this occupation probability changes with time? We can associate the change of occupation probability with a transfer rate :
(55)
The time evolution of the initial state can be described using the propagator
(56)
according to
(57)
To proceed, we form matrix elements of with the diabatic states and
(58)
These transition amplitudes are a measure of how much of the initial state is still contained in the propagated state . The time derivative of the transition amplitudes read
(59)
which leads to
(60)
Fourier transforming the last equation to frequency space, we obtain
(61)
Now, let us consider the special case of i.e., the survival amplitude of state
(62)
for which we obviously need to know
(63)
where we introduced an infinitesimal imaginary part to allow inversion of the equation. We will take the limit later. So, finally, we have found a closed expression for the survival amplitude
(64)
Now, consider
(65)
where denotes the Principal value. The timedependent occupation probability of the initial state is then determined as
(66)
To evaluate this integral, we make the following assumptions:
 The variation of and is weak for .
 The dependence results mainly from the resonance in .
Then, we can use the residue theorem to finally obtain
(67)
Taking the time derivative
(68)
and comparing this to Eq.55 one can read off the transfer rate
(69)
in which the term ensures energy conservation. For a set of final states instead of a single final states, the rate generalizes to Fermi’s Golden Rule
(70)
I.5 Harmonic approximation for nuclear motion
Assume that we have solved the electronic problem for different nuclear conformations, i.e., we have determined the potential energy surface (or the effective potential) for the nuclear dynamics. If the molecule exists, there is a stable equilibrium configuration with and we can express the dynamical nuclear coordinates using deviations from this configuration
(71)
We can then Taylor expand the effective potential around (nucleus , Cartesian coordinate )
(72)
In this harmonic approximation, the effective potential for the nuclear dynamics is given by
(73)
and consequently the nuclear Hamiltonian by
(74)
Introducing massweighted normal coordinates
(75)
one can rewrite the Hamiltonian as
(76)
where
(77)
is the dynamical matrix. This matrix can be diagonalized by an orthogonal transformation
(78)
in which the are normalmode frequencies. Applying the transformation to the coordinates and impulses
(79)
finally yields the nuclear Hamiltonian in the form
(80)
This Hamiltonian is evidently a sum over independent harmonic oscillators, for each of which the solution is known, i.e., is the eigenfunction of the harmonic oscillator and the total nuclear wave function is given by
(81)
The associated total energy is simply the sum over all individual energies
(82)