MODELING MATTER AT NANOSCALES 6.The theory of molecular orbitals for the description of nanosystems (part II) The density matrix.

Electron density from multielectronic wave functions The case of N identical antisymmetrized particles (fermions, as electrons are), characterized in the configuration space by  1,  2,..  N spatial and spin coordinates each, moving under the influence of a framework of fixed potentials (as their mutual interactions and those with nuclei) is the typical quantum system where nanoscopic phenomena and structures occur… 2

Electron density from multielectronic wave functions The case of N identical antisymmetrized particles (fermions, as electrons are), characterized in the configuration space by  1,  2,..  N spatial and spin coordinates each, moving under the influence of a framework of fixed potentials (as their mutual interactions and those with nuclei) is the typical quantum system where nanoscopic phenomena and structures occur… 3 where: is a permutation operator working on subindexes of each of the N coordinates p is the “parity” of each permutation (1 or 2) The wave function of a certain I state of such system can be given by:

Electron density from multielectronic wave functions Let us consider that: 4 is a volume differential of the configuration space regarding spatial and spin coordinates and allowing integration on all coordinates of all and each one of the electrons

Electron density from multielectronic wave functions A simple planar representation of the configuration space, where all electrons are placed at their respective coordinates, gives the idea that isolating a portion corresponding to a certain coordinate permits to evaluate how the whole system behaves in such portion:  2,  3, …,  N 11

Electron density from multielectronic wave functions To integrate the electron population in the configuration space we adopt a convention regarding particle coordinate differentials, and will agree that: is a notation for volume differentials of the remaining space after pointing to certain reference particle coordinate volume elements in positions  1,  2,... of interest.

Electron density from multielectronic wave functions To integrate the electron population in the configuration space we adopt a convention regarding particle coordinate differentials, and will agree that: is a notation for volume differentials of the remaining space after pointing to certain reference particle coordinate volume elements in positions  1,  2,... of interest. It means that refers to the volume differential of the entire electron system, except those of  l,  m,… point coordinates.

Electron density from multielectronic wave functions From quantum mechanical principles, it is well known that if  I is a normalized wave function of a multielectronic system in the I state, then: 8 means the probability that electron 1 has to be inside the volume and spatial element d  1 with coordinates  1, while electron 2 is at the same time in volume d  2 with coordinates  2, and so on for the entire system of N particles.

Electron density from multielectronic wave functions From quantum mechanical principles, it is well known that if  I is a normalized wave function of a multielectronic system in the I state, then: means the probability that electron 1 has to be inside the volume and spatial element d  1 with coordinates  1, while electron 2 is at the same time in volume d  2 with coordinates  2, and so on for the entire system of N particles. The integral of this product in the whole space is normalized to 1 The integral of this product in the whole space is normalized to 1.

Density matrices If we are only interested on the electron population at the  1 coordinate then: 10 is the probability to find a portion of all N electrons at the  1 spatial and spin coordinates of the configuration space corresponding to any I state of the system.

Density matrices If we are only interested on the electron population at the  1 coordinate then: 11 is the probability to find a portion of all N electrons at the  1 spatial and spin coordinates of the configuration space corresponding to any I state of the system. reduced density function  (  l ) is the reduced density function at the  l coordinate of the configuration space in an N – electron system, and is expressed as a non – dimensional number per unit of the spatial – spin “volume”.

Density matrices If we are only interested on the electron population at the  1 coordinate then: 12 is the probability to find a portion of all N electrons at the  1 spatial and spin coordinates of the configuration space corresponding to any I state of the system. is the non – dimensional number giving the electron population in the d  l volume differential at  l coordinate of the Hilbert space. Therefore: reduced density function  (  l ) is the reduced density function at the  l coordinate of the configuration space in an N – electron system, and is expressed as a non – dimensional number per unit of the spatial – spin “volume”.

Density matrices generalized density matrix of one particle The generalized density matrix of one particle is then given by: reduced density matrix of first order where  1 and  1 ´ are different coordinates in the configuration space corresponding to the subset of one particle. It is also known as the reduced density matrix of first order.

Density matrices generalized density matrix of one particle The generalized density matrix of one particle is then given by: reduced density matrix of first order where  1 and  1 ´ are different coordinates in the configuration space corresponding to the subset of one particle. It is also known as the reduced density matrix of first order. It means a mapping of density originated by the one particle portion of the Hilbert space of the whole system.

Density matrices generalized density matrix of one particle The generalized density matrix of one particle is then given by: reduced density matrix of first order where  1 and  1 ´ are different coordinates in the configuration space corresponding to the subset of one particle. It is also known as the reduced density matrix of first order. In this conditions the previously defined probability term  (  l ) is the diagonal element of this matrix for each  l coordinate. It means a mapping of density originated by the one particle portion of the Hilbert space of the whole system.

Density matrices reduced density functionof two interacting particles In a similar way, the reduced density function of two interacting particles can be defined as: It is the probability to find a portion of all N electrons at two  1 and  2 spatial and spin coordinates anywhere in the configuration space corresponding to a certain I state of the system.

Density matrices reduced density functionof two interacting particles In a similar way, the reduced density function of two interacting particles can be defined as: expresses the pair population of particles in the d  l d  m differential volume of such space at coordinates  l and  m. Therefore: It is the probability to find a portion of all N electrons at two  1 and  2 spatial and spin coordinates anywhere in the configuration space corresponding to a certain I state of the system.

Density matrices reduced density functionof two interacting particles In a similar way, the reduced density function of two interacting particles can be defined as: expresses the pair population of particles in the d  l d  m differential volume of such space at coordinates  l and  m. Therefore: is the number of order M sets in the total of N elements. It is the probability to find a portion of all N electrons at two  1 and  2 spatial and spin coordinates anywhere in the configuration space corresponding to a certain I state of the system.

Density matrices generalized density matrix of two particles The generalized density matrix of two particles is given by: reduced density matrix of second order where  1,  1 ´,  2,  2 ´ are different coordinates in the configuration space corresponding to the subset of particles 1 and 2. It is also known as the reduced density matrix of second order.

Density matrices generalized density matrix of two particles The generalized density matrix of two particles is given by: reduced density matrix of second order where  1,  1 ´,  2,  2 ´ are different coordinates in the configuration space corresponding to the subset of particles 1 and 2. It is also known as the reduced density matrix of second order. It means a mapping of the given pair of particles in the considered portion of the Hilbert space of the system.

Density matrices generalized density matrix of two particles The generalized density matrix of two particles is given by: reduced density matrix of second order where  1,  1 ´,  2,  2 ´ are different coordinates in the configuration space corresponding to the subset of particles 1 and 2. It is also known as the reduced density matrix of second order. In this conditions the previously defined probability term  (  l,  m ) is the diagonal element of this matrix for a given pair of particles. It means a mapping of the given pair of particles in the considered portion of the Hilbert space of the system.

Density matrices It can be proof that if two or more indices are equal, the matrix collapses to 0, because the particle antisymmetrization. 22

Density matrices It can be proof that if two or more indices are equal, the matrix collapses to 0, because the particle antisymmetrization. 23 It is the case of diagonal elements of two particle density matrices:

Density matrices It can be proof that if two or more indices are equal, the matrix collapses to 0, because the particle antisymmetrization. 24 Fermi’s hole Two electrons can not hold the same coordinates and it corresponds to the physical evidence that two Fermions with the same parallel spins can not coexist at small distances. It is the so called “Fermi’s hole”. It is the case of diagonal elements of two particle density matrices:

Density matrices All these expressions of any order p : 25 are two dimensional matrices where one function give columns and the conjugate the rows, or vice versa.

Density matrices As a consequence: 26 is the number of electrons because  is normalized to 1.

Density matrices As a consequence: 27 is the number of electron pairs. is the number of electrons because  is normalized to 1. Similarly:

Quantum properties depending on density matrices Any kind of quantum operator, being symmetrical with respect to particle indexes can be developed as: 28 corresponding ( 0 ), ( l ), ( lm ), ( lmn ),… to operate on zero, one, two, three,… particles, respectively.

Quantum properties depending on density matrices Any kind of quantum operator, being symmetrical with respect to particle indexes can be developed as: 29 corresponding ( 0 ), ( l ), ( lm ), ( lmn ),… to operate on zero, one, two, three,… particles, respectively. Primes indicate that terms with two or more identical subindex are excluded.

Quantum properties depending on density matrices In the case of two particles, taken as an example, the expectation value ã 1,2 corresponding to the value of property A related to their mutual interaction in the given I state of the system is: 30

Quantum properties depending on density matrices In the case of two particles, taken as an example, the expectation value ã 1,2 corresponding to the value of property A related to their mutual interaction in the given I state of the system is: 31 It must be observed that this transformation can only be made for cases where  1 =  1 ’ and  2 =  2 ‘.

Quantum properties depending on density matrices 32 The physical expectation value of the property A at the I state of the system can be obtained from, and de facto depends on, the diagonal of the corresponding generalized density matrix.

Quantum properties depending on density matrices In general, the expectation value ã corresponding to the value of property A in the given I state of the system in terms of the generalized density matrices is: 33

Quantum properties depending on density matrices In the case of a transition from state I to any state J of the system, the matrix element related with the change of property A in terms of the generalized density matrices is: 34

Quantum properties depending on density matrices Therefore, for evaluating a transition between two quantum states  I and  J of a multi – particle system we can define the transition matrix densities as: 35

Density matrices in terms of molecular orbitals In the special case of a ground state  0 that is a solution of a Hartree – Fock problem, the generalized density matrix of one l particle can be expressed in terms of molecular spin – orbitals of the Slater determinants and it remains as: 36 where the sum runs over all spin orbitals expressed in  0 and n i is their corresponding electron occupation.

Density matrices in terms of molecular orbitals In the special case of a ground state  0 that is a solution of a Hartree – Fock problem, the generalized density matrix of one l particle can be expressed in terms of molecular spin – orbitals of the Slater determinants and it remains as: 37 As the molecular orbitals are orthonormal, it holds again that: where the sum runs over all spin orbitals expressed in  0 and n i is their corresponding electron occupation.

Density matrices in terms of molecular orbitals Expressing the spatial component of MO’s in terms of a complete orthonormal atomic basis set:

Density matrices in terms of molecular orbitals Expressing the spatial component of MO’s in terms of a complete orthonormal atomic basis set: and performing the appropriate substitutions:

Energy and density functions Being the p density matrix element: 40

Energy and density functions Being the p density matrix element: 41 The expression of energy of the system remains as:

Energy and density functions Then the density function depends on the HF density matrix p as: 42

Energy and density functions Then the density function depends on the HF density matrix p as: 43 The density function is the trace of the Hartree – Fock density matrix in terms of an orthonormal basis of atomic orbitals.

Energy and density functions Then the density function depends on the HF density matrix p as: 44 The density function is the trace of the Hartree – Fock density matrix in terms of an orthonormal basis of atomic orbitals. This expression is very useful in cases when a given quantum state can not be expressed in terms of a wave function and it is in terms of a basis to define densities.

Energy and density functions Consequently: 45 If the density matrix of a system is available it can be obtained the expectation value of the energy, as well as it was developed in terms of a complete orthonormal basis set.

An application to dipole moments The operator for an electrical moment in a multielectronic system could be expressed as: 46

An application to dipole moments The operator for an electrical moment in a multielectronic system could be expressed as: 47 remains that the expectation value of the electric or dipole moment is only depending from the generalized density matrix of first order: When substituted in the previous expectation value formula: because only one particle terms have sense.

An application to dipole moments Consequently, the electric or dipole moment of transition between two states I and J can be written as: 48

An application to dipole moments Consequently, the electric or dipole moment of transition between two states I and J can be written as: 49 In the case of dealing with electronic states of nanoscopic systems becomes evident that a change in the distribution of the multielectronic cloud means changing the quantum state of the system.