MODELING MATTER AT NANOSCALES 4 Introdution to quantum treatments 403 Eigenvetors and eigenvalues of a matrix
Simultaneous equations in the variational method The problem of simultaneous equations in the variational method an be written as: HC SEC = 0 The solution an be ahieved when the referene funtions to the arbitrary solution are orthonormal In that ase the overlap matrix beomes a unitary δ mn = 1, and the expression beomes: HC 1EC = 0
Simultaneous equations in the variational method The problem of simultaneous equations in the variational method an be written as: HC SEC = 0 The solution an be ahieved when the referene funtions to the arbitrary solution are orthonormal In that ase the overlap matrix beomes a unitary δ mn = 1, and the expression beomes: HC 1EC = 0
Simultaneous equations in the variational method The problem of simultaneous equations in the variational method an be written as: HC SEC = 0 The solution an be ahieved when the referene funtions to the arbitrary solution are orthonormal In that ase the overlap matrix beomes a unitary δ mn = 1, and the expression beomes: HC 1EC = 0
Simultaneous equations in the variational method Identity matrix an be omitted and then: HC EC = 0 HC = EC H = CEC -1 The determinantal equation orresponding to the orthonormal referene is known as the system s seular equation: H E = 0
Simultaneous equations in the variational method Identity matrix an be omitted and then: HC EC = 0 HC = EC H = CEC -1 The determinantal equation orresponding to the orthonormal referene is known as the system s seular equation: H E = 0
Diagonalization A given matrix H beomes diagonalized when both a diagonal matrix E and a reversible matrix C exists and are found, suh that: H = CEC -1 It relates the problem of obtaining expeted or exat values for the Shrödinger equation if H is the Hamiltonian matrix and C orresponds to oeffiients of partiipation in the linear ombination of system s referene wave funtions
Diagonalization A given matrix H beomes diagonalized when both a diagonal matrix E and a reversible matrix C exists and are found, suh that: H = CEC -1 It relates the problem of obtaining expeted or exat values for the Shrödinger equation if H is the Hamiltonian matrix and C orresponds to oeffiients of partiipation in the linear ombination of system s referene wave funtions
Diagonalization Diagonalization of H matrix an mostly be performed when it is symmetri (H µν = H νµ ) It is a routine treatment of numeri mathematis Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matries
Diagonalization Diagonalization of H matrix an mostly be performed when it is symmetri (H µν = H νµ ) It is a routine treatment of numeri mathematis Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matries
Diagonalization Diagonalization of H matrix an mostly be performed when it is symmetri (H µν = H νµ ) It is a routine treatment of numeri mathematis Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matries
Eigenvalues and eigenvetors Diagonalization is also known as the ase of finding eigenvalues and eigenvetors of a symmetri matrix Eigenvalues are the E k terms of the diagonal E matrix Eigenvetors are given by the iµ oeffiients in C matrix
Eigenvalues and eigenvetors Diagonalization is also known as the ase of finding eigenvalues and eigenvetors of a symmetri matrix Eigenvalues are the E k terms of the diagonal E matrix Eigenvetors are given by the iµ oeffiients in C matrix
Eigenvalues and eigenvetors Diagonalization is also known as the ase of finding eigenvalues and eigenvetors of a symmetri matrix Eigenvalues are the E k terms of the diagonal E matrix Eigenvetors are given by the iµ oeffiients in C matrix
Eigenvalues and eigenvetors Eigenvetors resulting from routine diagonalization proedures are orthonormal: = 1 C matries are non-symmetri i i 2 iµ iµ iν = 0
Eigenvalues and eigenvetors Eigenvetors resulting from routine diagonalization proedures are orthonormal: = 1 C matries are non-symmetri i i 2 iµ iµ iν = 0
Orbitals as a ase of referene
MO as LCAO One of the most relevant appliations of linear algebra for understanding the nanosopi universe is the onsideration of wave funtions as referenes that an enter variational optimizations for building solutions to the state wave funtion of omplex systems Suh referene wave funtions usually orrespond to the state of a single eletron, either in an atom, a bond, a moleule, a unit rystal, et and are defined as orbitals
MO as LCAO One of the most relevant appliations of linear algebra for understanding the nanosopi universe is the onsideration of wave funtions as referenes that an enter variational optimizations for building solutions to the state wave funtion of omplex systems Suh referene wave funtions usually orrespond to the state of a single eletron, either in an atom, a bond, a moleule, a unit rystal, et and are defined as orbitals
MO as LCAO In order to optimize one-eletron wave funtions of moleules, or Moleular Orbitals (MO), the Linear Combination of Atomi Orbitals (LCAO) has often been used: ψ i = φ + φ + i1 1 i2 = iµφ µ where oeffiients iµ provide the partiipation of the basis or referene atomi orbital φ µ in the state or moleular orbital ψ i 2 µ
MO as LCAO In order to optimize one-eletron wave funtions of moleules, or Moleular Orbitals (MO), the Linear Combination of Atomi Orbitals (LCAO) has often been used: ψ i = φ + φ + i1 1 i2 = iµφ µ where oeffiients iµ provide the partiipation of the basis or referene atomi orbital φ µ in the state or moleular orbital ψ i 2 µ
MO as LCAO If the oeffiient or eigenvetor matrix is onsidered as: and the system s matrix is given by energies in terms of the atomi orbital struture is given as: C = 11 21 i i 1 12 22 1µ µ Then, a produt an be obtained as: HC = EC
MO as LCAO If the oeffiient or eigenvetor matrix is onsidered as: and the system s matrix is given by energies in terms of the atomi orbital struture is given as: C = H = 11 21 i i H H 1 11 21 H H 12 22 12 22 1µ H µ 1µ H i 1 H i µ Then, a produt an be obtained as: HC = EC
MO as LCAO If the oeffiient or eigenvetor matrix is onsidered as: and the system s matrix is given by energies in terms of the atomi orbital struture is given as: C = H = 11 21 i i H H 1 11 21 H H 12 22 12 22 1µ H µ 1µ H i 1 H i µ Then, a produt an be obtained as: HC = EC
MO as LCAO E is then a diagonal matrix terming E k eigenvalues of all onsidered one-eletron wave funtions of the system: E = E 0 0 1 0 E 2 0 E µ
MO as LCAO E an be obtained by the oeffiient C matrix from H after a linear transformation, or diagonalization: E = CHC -1
MO as LCAO H is onstruted by energies of eah basis set omponent (basis orbitals) in the system s ontext E is a result of a linear transformation of H energy matrix by C eigenvetor matrix of MO s oeffiients giving the minimal total energy This step is onsidered as the leitmotif of almost all quantum mehanial alulations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the omplete system
MO as LCAO H is onstruted by energies of eah basis set omponent (basis orbitals) in the system s ontext E is a result of a linear transformation of H energy matrix by C eigenvetor matrix of MO s oeffiients giving the minimal total energy This step is onsidered as the leitmotif of almost all quantum mehanial alulations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the omplete system
MO as LCAO H is onstruted by energies of eah basis set omponent (basis orbitals) in the system s ontext E is a result of a linear transformation of H energy matrix by C eigenvetor matrix of MO s oeffiients giving the minimal total energy This step is onsidered as the leitmotif of almost all quantum mehanial alulations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the omplete system