From 09fe59d4e92958be790af84294c9fe388057192e Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 20 Sep 2024 07:11:14 +0200 Subject: [PATCH] update --- doc/pub/week38/html/week38-bs.html | 397 ++-------- doc/pub/week38/html/week38-reveal.html | 324 ++------ doc/pub/week38/html/week38-solarized.html | 360 ++------- doc/pub/week38/html/week38.html | 360 ++------- doc/pub/week38/ipynb/ipynb-week38-src.tar.gz | Bin 193 -> 193 bytes doc/pub/week38/ipynb/week38.ipynb | 782 ++++++------------- doc/pub/week38/pdf/week38.pdf | Bin 327759 -> 302712 bytes doc/src/week38/week38.do.txt | 258 ++---- 8 files changed, 603 insertions(+), 1878 deletions(-) diff --git a/doc/pub/week38/html/week38-bs.html b/doc/pub/week38/html/week38-bs.html index 4db63dac..0dbd1f8e 100644 --- a/doc/pub/week38/html/week38-bs.html +++ b/doc/pub/week38/html/week38-bs.html @@ -107,54 +107,35 @@ 2, None, 'creation-operators-in-terms-of-pauli-matrices'), - ('Full configuration interaction theory', + ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Slater determinants as basis states, Repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states, repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states', - 2, - None, - 'slater-determinants-as-basis-states'), - ('Slater determinants as basis states', + ('One-particle-one-hole state', 2, None, - 'slater-determinants-as-basis-states'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), + 'one-particle-one-hole-state'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Full Configuration Interaction Theory', + ('Intermediate normalization', 2, None, - 'full-configuration-interaction-theory'), + 'intermediate-normalization'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Compact expression of correlated part', + 2, + None, + 'compact-expression-of-correlated-part'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Minimization', 2, None, 'minimization'), ('Full Configuration Interaction Theory', 2, None, @@ -172,6 +153,10 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Using the Condon-Slater rule', + 2, + None, + 'using-the-condon-slater-rule'), ('A non-practical way of solving the eigenvalue problem', 2, None, @@ -180,6 +165,7 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Slight rewrite', 2, None, 'slight-rewrite'), ('Rewriting the FCI equation', 2, None, @@ -192,10 +178,11 @@ 2, None, 'rewriting-the-fci-equation-does-not-stop-here'), - ('Rewriting the FCI equation, please stop here', + ('Finding the coefficients', 2, None, 'finding-the-coefficients'), + ('Rewriting the FCI equation', 2, None, - 'rewriting-the-fci-equation-please-stop-here'), + 'rewriting-the-fci-equation'), ('Rewriting the FCI equation, more to add', 2, None, @@ -211,7 +198,8 @@ ('Definition of the correlation energy', 2, None, - 'definition-of-the-correlation-energy')]} + 'definition-of-the-correlation-energy'), + ('Ground state energy', 2, None, 'ground-state-energy')]} end of tocinfo --> @@ -275,33 +263,33 @@
  • Raising and lowewring matrices
  • Transformation of operators
  • Creation operators in terms of Pauli matrices
    • -
  • Full configuration interaction theory
    • -
  • Slater determinants as basis states, Repetition
    • -
  • Slater determinants as basis states, repetition
    • -
  • Slater determinants as basis states
    • -
  • Slater determinants as basis states
    • -
  • Quick repetition of the occupation representation
    • -
  • Quick repetition of the occupation representation
    • -
  • Quick repetition of the occupation representation
  • Full Configuration Interaction Theory
    • +
  • One-particle-one-hole state
  • Full Configuration Interaction Theory
    • +
  • Intermediate normalization
  • Full Configuration Interaction Theory
    • +
  • Compact expression of correlated part
  • Full Configuration Interaction Theory
    • +
  • Minimization
  • Full Configuration Interaction Theory
  • Full Configuration Interaction Theory
  • FCI and the exponential growth
  • Exponential wall
  • A non-practical way of solving the eigenvalue problem
    • +
  • Using the Condon-Slater rule
  • A non-practical way of solving the eigenvalue problem
  • A non-practical way of solving the eigenvalue problem
    • +
  • Slight rewrite
  • Rewriting the FCI equation
  • Rewriting the FCI equation
  • Rewriting the FCI equation, does not stop here
    • -
  • Rewriting the FCI equation, please stop here
    • +
  • Finding the coefficients
    • +
  • Rewriting the FCI equation
  • Rewriting the FCI equation, more to add
  • Rewriting the FCI equation, more to add
  • Summarizing FCI and bringing in approximative methods
  • Definition of the correlation energy
    • +
  • Ground state energy
    • @@ -943,206 +931,19 @@

    Creation o

    -

    Full configuration interaction theory

    - -

    We start with a reminder on determinants in the number representation.

    - - -

    Slater determinants as basis states, Repetition

    -
    -
    - -

    The simplest possible choice for many-body wavefunctions are product wavefunctions. -That is -

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots -$$ - -

    because we are really only good at thinking about one particle at a time. Such -product wavefunctions, without correlations, are easy to -work with; for example, if the single-particle states \( \phi_i(x) \) are orthonormal, then -the product wavefunctions are easy to orthonormalize. -

    - -

    Similarly, computing matrix elements of operators are relatively easy, because the -integrals factorize. -

    - -

    The price we pay is the lack of correlations, which we must build up by using many, many product -wavefunctions. (Thus we have a trade-off: compact representation of correlations but -difficult integrals versus easy integrals but many states required.) -

    -
    -
    - - - -

    Slater determinants as basis states, repetition

    -
    -
    - -

    Because we have fermions, we are required to have antisymmetric wavefunctions, e.g.

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) = - \Psi(x_2, x_1, x_3, \ldots, x_N) -$$ - -

    etc. This is accomplished formally by using the determinantal formalism

    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Product wavefunction + antisymmetry = Slater determinant.

    -
    -
    - - - -

    Slater determinants as basis states

    -
    -
    - -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Properties of the determinant (interchange of any two rows or -any two columns yields a change in sign; thus no two rows and no -two columns can be the same) lead to the Pauli principle: -

    - -
      -
    • No two particles can be at the same place (two columns the same); and
      • -
    • No two particles can be in the same state (two rows the same).
      • -
    -
    -
    - - - -

    Slater determinants as basis states

    -
    -
    - -

    As a practical matter, however, Slater determinants beyond \( N=4 \) quickly become -unwieldy. Thus we turn to the occupation representation or second quantization to simplify calculations. -

    - -

    The occupation representation or number representation, using fermion creation and annihilation -operators, is compact and efficient. It is also abstract and, at first encounter, not easy to -internalize. It is inspired by other operator formalism, such as the ladder operators for -the harmonic oscillator or for angular momentum, but unlike those cases, the operators do not have coordinate space representations. -

    - -

    Instead, one can think of fermion creation/annihilation operators as a game of symbols that -compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater -determinants. -

    -
    -
    - - - -

    Quick repetition of the occupation representation

    -
    -
    - -

    We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). -(Note: this requirement, and others, can be relaxed, but leads to a -more involved formalism.) Any orthonormal set will do. -

    - -

    To each single-particle state \( \phi_i(x) \) we associate a creation operator -\( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \). -

    - -

    When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes -a particle to occupy the single-particle state \( \phi_i(x) \): -

    -$$ -\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle -$$ -
    -
    - - - -

    Quick repetition of the occupation representation

    -
    -
    - -

    But with multiple creation operators we can occupy multiple states:

    -$$ -\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime}) -\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle. -$$ - -

    Now we impose antisymmetry, by having the fermion operators satisfy anticommutation relations:

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i -= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+ -= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0 -$$ - -

    so that

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i -$$ -
    -
    - +

    Full Configuration Interaction Theory

    - -

    Quick repetition of the occupation representation

    -
    -
    - -

    Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), -enforcing the Pauli exclusion principle. Thus when writing a Slater determinant -using creation operators, -

    +

    We have defined the ansatz for the ground state as

    $$ -\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle +|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, $$ -

    each index \( i,j,k, \ldots \) must be unique.

    - -

    For some relevant exercises with solutions see chapter 8 of Lecture Notes in Physics, volume 936.

    -
    -
    - +

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions.

    -

    Full Configuration Interaction Theory

    -
    -
    - -

    We have defined the ansatz for the ground state as

    -$$ -|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, -$$ +

    One-particle-one-hole state

    -

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions. -A given one-particle-one-hole (\( 1p1h \)) state can be written as -

    +

    A given one-particle-one-hole (\( 1p1h \)) state can be written as

    $$ |\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle, $$ @@ -1156,15 +957,11 @@

    Full Configuration $$ |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. $$ -

    -

    Full Configuration Interaction Theory

    -
    -
    - +

    We can then expand our exact state function for the ground state as

    @@ -1178,6 +975,9 @@

    Full Configuration \hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots $$ + + +

    Intermediate normalization

    Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have

    @@ -1189,15 +989,11 @@

    Full Configuration $$ |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. $$ -

    -

    Full Configuration Interaction Theory

    -
    -
    - +

    We rewrite

    $$ |\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots, @@ -1208,9 +1004,12 @@

    Full Configuration |\Psi_0\rangle=\sum_{PH}C_H^P\Phi_H^P=\left(\sum_{PH}C_H^P\hat{A}_H^P\right)|\Phi_0\rangle, $$ -

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states. -We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators. -

    +

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states.

    + + +

    Compact expression of correlated part

    + +

    We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators.

    Our requirement of unit normalization gives

    $$ @@ -1221,15 +1020,11 @@

    Full Configuration $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. $$ -

    -

    Full Configuration Interaction Theory

    -
    -
    - +

    Normally

    $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}, @@ -1242,9 +1037,12 @@

    Full Configuration \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle, $$ -

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system. -The minimization process results in -

    +

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system.

    + + +

    Minimization

    + +

    The minimization process results in

    $$ \delta\left[ \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle\right]=0, $$ @@ -1254,16 +1052,10 @@

    Full Configuration \sum_{P'H'}\left\{\delta[C_H^{*P}]\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}- \lambda( \delta[C_H^{*P}]C_{H'}^{P'}]\right\} = 0. $$ -

    -

    Full Configuration Interaction Theory

    -
    -
    - -

    This leads to

    $$ \sum_{P'H'}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda C_H^{P}=0, @@ -1277,15 +1069,10 @@

    Full Configuration $$

    leading to the identification \( \lambda = E \).

    -

    -
    -

    Full Configuration Interaction Theory

    -
    -
    - +

    An alternative way to derive the last equation is to start from

    $$ (\hat{H} -E)|\Psi_0\rangle = (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0, @@ -1297,15 +1084,10 @@

    Full Configuration of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator \( \hat{C} \).

    -

    -
    -

    FCI and the exponential growth

    -
    -
    - +

    Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.

    The total number of Slater determinants which can be built with say \( N \) neutrons distributed among \( n \) single particle states is

    @@ -1319,9 +1101,6 @@

    FCI and the exponential g $$

    and multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality \( d \) of \( d\sim 10^{18} \).

    -

    -
    -

    Exponential wall

    @@ -1340,9 +1119,7 @@

    Exponential wall

    A non-practical way of solving the eigenvalue problem

    -
    -
    - +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1355,7 +1132,10 @@

    A (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the + + +

    Using the Condon-Slater rule

    +

    If we assume that we have a two-body operator at most, using the Condon-Slater rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ @@ -1372,15 +1152,10 @@

    A

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -
    -

    A non-practical way of solving the eigenvalue problem

    -
    -
    - +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1392,24 +1167,24 @@

    A $$ (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    -

    A non-practical way of solving the eigenvalue problem

    -
    -
    - +

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ \langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+ -\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0, +\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0. $$ -

    or

    + + +

    Slight rewrite

    + +

    Which we can rewrite

    $$ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}, @@ -1418,17 +1193,14 @@

    A

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -
    -

    Rewriting the FCI equation

    -

    In our notes on Hartree-Fock calculations, -we have already computed the matrix \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then the matrix elements +

    In our discussions of the Hartree-Fock method planned for week 39, +we are going to compute the elements \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then these quantities result in \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and we are left with a correlation energy given by

    $$ @@ -1456,28 +1228,27 @@

    Rewriting the FCI equation <

    Rewriting the FCI equation, does not stop here

    -
    -
    - +

    We need more equations. Our next step is to set up

    $$ \langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ -\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0, +\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0. $$ -

    as this equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    + + +

    Finding the coefficients

    +

    This equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    $$ \langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ \sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a. $$ -
    -
    -

    Rewriting the FCI equation, please stop here

    +

    Rewriting the FCI equation

    @@ -1554,18 +1325,18 @@

    Su

    Definition of the correlation energy

    -
    -
    - +

    The correlation energy is defined as, with a two-body Hamiltonian,

    $$ \Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+ \sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}. $$ -

    The coefficients \( C \) result from the solution of the eigenvalue problem. -The energy of say the ground state is then -

    +

    The coefficients \( C \) result from the solution of the eigenvalue problem.

    + + +

    Ground state energy

    +

    The energy of say the ground state is then

    $$ E=E_{ref}+\Delta E, $$ @@ -1574,8 +1345,6 @@

    Definition of the c $$ E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle. $$ -

    -
    diff --git a/doc/pub/week38/html/week38-reveal.html b/doc/pub/week38/html/week38-reveal.html index 0d3905ff..b1c199c7 100644 --- a/doc/pub/week38/html/week38-reveal.html +++ b/doc/pub/week38/html/week38-reveal.html @@ -901,220 +901,22 @@

    Creation operators in ter
    -

    Full configuration interaction theory

    - -

    We start with a reminder on determinants in the number representation.

    -
    - -
    -

    Slater determinants as basis states, Repetition

    -
    - -

    -

    The simplest possible choice for many-body wavefunctions are product wavefunctions. -That is -

    -

     
    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots -$$ -

     
    - -

    because we are really only good at thinking about one particle at a time. Such -product wavefunctions, without correlations, are easy to -work with; for example, if the single-particle states \( \phi_i(x) \) are orthonormal, then -the product wavefunctions are easy to orthonormalize. -

    - -

    Similarly, computing matrix elements of operators are relatively easy, because the -integrals factorize. -

    - -

    The price we pay is the lack of correlations, which we must build up by using many, many product -wavefunctions. (Thus we have a trade-off: compact representation of correlations but -difficult integrals versus easy integrals but many states required.) -

    -
    -
    - -
    -

    Slater determinants as basis states, repetition

    -
    - -

    -

    Because we have fermions, we are required to have antisymmetric wavefunctions, e.g.

    -

     
    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) = - \Psi(x_2, x_1, x_3, \ldots, x_N) -$$ -

     
    - -

    etc. This is accomplished formally by using the determinantal formalism

    -

     
    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ -

     
    - -

    Product wavefunction + antisymmetry = Slater determinant.

    -
    -
    - -
    -

    Slater determinants as basis states

    -
    - -

    -

     
    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ -

     
    - -

    Properties of the determinant (interchange of any two rows or -any two columns yields a change in sign; thus no two rows and no -two columns can be the same) lead to the Pauli principle: -

    - -
      -

    • No two particles can be at the same place (two columns the same); and
      • -

    • No two particles can be in the same state (two rows the same).
      • -
    -
    -
    - -
    -

    Slater determinants as basis states

    -
    - -

    -

    As a practical matter, however, Slater determinants beyond \( N=4 \) quickly become -unwieldy. Thus we turn to the occupation representation or second quantization to simplify calculations. -

    - -

    The occupation representation or number representation, using fermion creation and annihilation -operators, is compact and efficient. It is also abstract and, at first encounter, not easy to -internalize. It is inspired by other operator formalism, such as the ladder operators for -the harmonic oscillator or for angular momentum, but unlike those cases, the operators do not have coordinate space representations. -

    - -

    Instead, one can think of fermion creation/annihilation operators as a game of symbols that -compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater -determinants. -

    -
    -
    - -
    -

    Quick repetition of the occupation representation

    -
    - -

    -

    We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). -(Note: this requirement, and others, can be relaxed, but leads to a -more involved formalism.) Any orthonormal set will do. -

    - -

    To each single-particle state \( \phi_i(x) \) we associate a creation operator -\( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \). -

    - -

    When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes -a particle to occupy the single-particle state \( \phi_i(x) \): -

    -

     
    -$$ -\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle -$$ -

     
    -

    -
    - -
    -

    Quick repetition of the occupation representation

    -
    - -

    -

    But with multiple creation operators we can occupy multiple states:

    -

     
    -$$ -\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime}) -\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle. -$$ -

     
    - -

    Now we impose antisymmetry, by having the fermion operators satisfy anticommutation relations:

    -

     
    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i -= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+ -= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0 -$$ -

     
    - -

    so that

    -

     
    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i -$$ -

     
    -

    -
    +

    Full Configuration Interaction Theory

    -
    -

    Quick repetition of the occupation representation

    -
    - -

    -

    Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), -enforcing the Pauli exclusion principle. Thus when writing a Slater determinant -using creation operators, -

    +

    We have defined the ansatz for the ground state as

     
    $$ -\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle +|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, $$

     
    -

    each index \( i,j,k, \ldots \) must be unique.

    - -

    For some relevant exercises with solutions see chapter 8 of Lecture Notes in Physics, volume 936.

    -
    +

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions.

    -

    Full Configuration Interaction Theory

    -
    - -

    -

    We have defined the ansatz for the ground state as

    -

     
    -$$ -|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, -$$ -

     
    +

    One-particle-one-hole state

    -

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions. -A given one-particle-one-hole (\( 1p1h \)) state can be written as -

    +

    A given one-particle-one-hole (\( 1p1h \)) state can be written as

     
    $$ |\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle, @@ -1134,14 +936,11 @@

    Full Configuration Interaction Th |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. $$

     
    -

    Full Configuration Interaction Theory

    -
    - -

    +

    We can then expand our exact state function for the ground state as

    @@ -1158,7 +957,10 @@

    Full Configuration Interaction Th \hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots $$

     
    +

    +
    +

    Intermediate normalization

    Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have

    @@ -1174,14 +976,11 @@

    Full Configuration Interaction Th |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. $$

     
    -

    Full Configuration Interaction Theory

    -
    - -

    +

    We rewrite

     
    $$ @@ -1196,9 +995,13 @@

    Full Configuration Interaction Th $$

     
    -

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states. -We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators. -

    +

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states.

    +

    + +
    +

    Compact expression of correlated part

    + +

    We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators.

    Our requirement of unit normalization gives

     
    @@ -1213,14 +1016,11 @@

    Full Configuration Interaction Th E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. $$

     
    -

    Full Configuration Interaction Theory

    -
    - -

    +

    Normally

     
    $$ @@ -1237,9 +1037,13 @@

    Full Configuration Interaction Th $$

     
    -

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system. -The minimization process results in -

    +

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system.

    +

    + +
    +

    Minimization

    + +

    The minimization process results in

     
    $$ \delta\left[ \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle\right]=0, @@ -1253,15 +1057,10 @@

    Full Configuration Interaction Th \lambda( \delta[C_H^{*P}]C_{H'}^{P'}]\right\} = 0. $$

     
    -

    Full Configuration Interaction Theory

    -
    - -

    -

    This leads to

     
    $$ @@ -1279,14 +1078,11 @@

    Full Configuration Interaction Th

     

    leading to the identification \( \lambda = E \).

    -

    Full Configuration Interaction Theory

    -
    - -

    +

    An alternative way to derive the last equation is to start from

     
    $$ @@ -1300,14 +1096,11 @@

    Full Configuration Interaction Th of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator \( \hat{C} \).

    -

    FCI and the exponential growth

    -
    - -

    +

    Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.

    The total number of Slater determinants which can be built with say \( N \) neutrons distributed among \( n \) single particle states is

    @@ -1325,7 +1118,6 @@

    FCI and the exponential growth

     

    and multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality \( d \) of \( d\sim 10^{18} \).

    -
    @@ -1344,9 +1136,7 @@

    Exponential wall

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

     
    $$ @@ -1362,8 +1152,11 @@

    A non-practical w (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$

     
    +

    -

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the +

    +

    Using the Condon-Slater rule

    +

    If we assume that we have a two-body operator at most, using the Condon-Slater rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

     
    @@ -1384,14 +1177,11 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

     
    $$ @@ -1407,25 +1197,26 @@

    A non-practical w (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$

     
    -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

     
    $$ \langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+ -\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0, +\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0. $$

     
    +

    -

    or

    +
    +

    Slight rewrite

    + +

    Which we can rewrite

     
    $$ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ @@ -1436,7 +1227,6 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    @@ -1444,8 +1234,8 @@

    Rewriting the FCI equation

    -

    In our notes on Hartree-Fock calculations, -we have already computed the matrix \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then the matrix elements +

    In our discussions of the Hartree-Fock method planned for week 39, +we are going to compute the elements \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then these quantities result in \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and we are left with a correlation energy given by

     
    @@ -1475,19 +1265,20 @@

    Rewriting the FCI equation

    Rewriting the FCI equation, does not stop here

    -
    - -

    +

    We need more equations. Our next step is to set up

     
    $$ \langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ -\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0, +\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0. $$

     
    +

    -

    as this equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    +
    +

    Finding the coefficients

    +

    This equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

     
    $$ \langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+ @@ -1495,11 +1286,10 @@

    Rewriting the FCI equatio \sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a. $$

     
    -

    -

    Rewriting the FCI equation, please stop here

    +

    Rewriting the FCI equation

    @@ -1580,9 +1370,7 @@

    Summarizing FCI a

    Definition of the correlation energy

    -
    - -

    +

    The correlation energy is defined as, with a two-body Hamiltonian,

     
    $$ @@ -1591,9 +1379,12 @@

    Definition of the correlation ener $$

     
    -

    The coefficients \( C \) result from the solution of the eigenvalue problem. -The energy of say the ground state is then -

    +

    The coefficients \( C \) result from the solution of the eigenvalue problem.

    +

    + +
    +

    Ground state energy

    +

    The energy of say the ground state is then

     
    $$ E=E_{ref}+\Delta E, @@ -1606,7 +1397,6 @@

    Definition of the correlation ener E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle. $$

     
    -

    diff --git a/doc/pub/week38/html/week38-solarized.html b/doc/pub/week38/html/week38-solarized.html index 7eba4d81..67264bfb 100644 --- a/doc/pub/week38/html/week38-solarized.html +++ b/doc/pub/week38/html/week38-solarized.html @@ -134,54 +134,35 @@ 2, None, 'creation-operators-in-terms-of-pauli-matrices'), - ('Full configuration interaction theory', + ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Slater determinants as basis states, Repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states, repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states', - 2, - None, - 'slater-determinants-as-basis-states'), - ('Slater determinants as basis states', + ('One-particle-one-hole state', 2, None, - 'slater-determinants-as-basis-states'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), + 'one-particle-one-hole-state'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Full Configuration Interaction Theory', + ('Intermediate normalization', 2, None, - 'full-configuration-interaction-theory'), + 'intermediate-normalization'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Compact expression of correlated part', + 2, + None, + 'compact-expression-of-correlated-part'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Minimization', 2, None, 'minimization'), ('Full Configuration Interaction Theory', 2, None, @@ -199,6 +180,10 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Using the Condon-Slater rule', + 2, + None, + 'using-the-condon-slater-rule'), ('A non-practical way of solving the eigenvalue problem', 2, None, @@ -207,6 +192,7 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Slight rewrite', 2, None, 'slight-rewrite'), ('Rewriting the FCI equation', 2, None, @@ -219,10 +205,11 @@ 2, None, 'rewriting-the-fci-equation-does-not-stop-here'), - ('Rewriting the FCI equation, please stop here', + ('Finding the coefficients', 2, None, 'finding-the-coefficients'), + ('Rewriting the FCI equation', 2, None, - 'rewriting-the-fci-equation-please-stop-here'), + 'rewriting-the-fci-equation'), ('Rewriting the FCI equation, more to add', 2, None, @@ -238,7 +225,8 @@ ('Definition of the correlation energy', 2, None, - 'definition-of-the-correlation-energy')]} + 'definition-of-the-correlation-energy'), + ('Ground state energy', 2, None, 'ground-state-energy')]} end of tocinfo --> @@ -886,199 +874,19 @@

    Creation operators in ter











    -

    Full configuration interaction theory

    - -

    We start with a reminder on determinants in the number representation.

    - -









    -

    Slater determinants as basis states, Repetition

    -
    - -

    -

    The simplest possible choice for many-body wavefunctions are product wavefunctions. -That is -

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots -$$ - -

    because we are really only good at thinking about one particle at a time. Such -product wavefunctions, without correlations, are easy to -work with; for example, if the single-particle states \( \phi_i(x) \) are orthonormal, then -the product wavefunctions are easy to orthonormalize. -

    - -

    Similarly, computing matrix elements of operators are relatively easy, because the -integrals factorize. -

    - -

    The price we pay is the lack of correlations, which we must build up by using many, many product -wavefunctions. (Thus we have a trade-off: compact representation of correlations but -difficult integrals versus easy integrals but many states required.) -

    -
    - - -









    -

    Slater determinants as basis states, repetition

    -
    - -

    -

    Because we have fermions, we are required to have antisymmetric wavefunctions, e.g.

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) = - \Psi(x_2, x_1, x_3, \ldots, x_N) -$$ - -

    etc. This is accomplished formally by using the determinantal formalism

    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Product wavefunction + antisymmetry = Slater determinant.

    -
    - - -









    -

    Slater determinants as basis states

    -
    - -

    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Properties of the determinant (interchange of any two rows or -any two columns yields a change in sign; thus no two rows and no -two columns can be the same) lead to the Pauli principle: -

    - - -
    - - -









    -

    Slater determinants as basis states

    -
    - -

    -

    As a practical matter, however, Slater determinants beyond \( N=4 \) quickly become -unwieldy. Thus we turn to the occupation representation or second quantization to simplify calculations. -

    - -

    The occupation representation or number representation, using fermion creation and annihilation -operators, is compact and efficient. It is also abstract and, at first encounter, not easy to -internalize. It is inspired by other operator formalism, such as the ladder operators for -the harmonic oscillator or for angular momentum, but unlike those cases, the operators do not have coordinate space representations. -

    - -

    Instead, one can think of fermion creation/annihilation operators as a game of symbols that -compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater -determinants. -

    -
    - - -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). -(Note: this requirement, and others, can be relaxed, but leads to a -more involved formalism.) Any orthonormal set will do. -

    - -

    To each single-particle state \( \phi_i(x) \) we associate a creation operator -\( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \). -

    - -

    When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes -a particle to occupy the single-particle state \( \phi_i(x) \): -

    -$$ -\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle -$$ -
    - - -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    But with multiple creation operators we can occupy multiple states:

    -$$ -\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime}) -\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle. -$$ - -

    Now we impose antisymmetry, by having the fermion operators satisfy anticommutation relations:

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i -= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+ -= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0 -$$ - -

    so that

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i -$$ -
    - +

    Full Configuration Interaction Theory

    -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), -enforcing the Pauli exclusion principle. Thus when writing a Slater determinant -using creation operators, -

    +

    We have defined the ansatz for the ground state as

    $$ -\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle +|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, $$ -

    each index \( i,j,k, \ldots \) must be unique.

    - -

    For some relevant exercises with solutions see chapter 8 of Lecture Notes in Physics, volume 936.

    -
    - +

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions.











    -

    Full Configuration Interaction Theory

    -
    - -

    -

    We have defined the ansatz for the ground state as

    -$$ -|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, -$$ +

    One-particle-one-hole state

    -

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions. -A given one-particle-one-hole (\( 1p1h \)) state can be written as -

    +

    A given one-particle-one-hole (\( 1p1h \)) state can be written as

    $$ |\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle, $$ @@ -1092,14 +900,11 @@

    Full Configuration Interaction Th $$ |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    We can then expand our exact state function for the ground state as

    @@ -1113,6 +918,9 @@

    Full Configuration Interaction Th \hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots $$ + +









    +

    Intermediate normalization

    Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have

    @@ -1124,14 +932,11 @@

    Full Configuration Interaction Th $$ |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    We rewrite

    $$ |\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots, @@ -1142,9 +947,12 @@

    Full Configuration Interaction Th |\Psi_0\rangle=\sum_{PH}C_H^P\Phi_H^P=\left(\sum_{PH}C_H^P\hat{A}_H^P\right)|\Phi_0\rangle, $$ -

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states. -We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators. -

    +

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states.

    + +









    +

    Compact expression of correlated part

    + +

    We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators.

    Our requirement of unit normalization gives

    $$ @@ -1155,14 +963,11 @@

    Full Configuration Interaction Th $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    Normally

    $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}, @@ -1175,9 +980,12 @@

    Full Configuration Interaction Th \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle, $$ -

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system. -The minimization process results in -

    +

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system.

    + +









    +

    Minimization

    + +

    The minimization process results in

    $$ \delta\left[ \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle\right]=0, $$ @@ -1187,15 +995,10 @@

    Full Configuration Interaction Th \sum_{P'H'}\left\{\delta[C_H^{*P}]\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}- \lambda( \delta[C_H^{*P}]C_{H'}^{P'}]\right\} = 0. $$ -











    Full Configuration Interaction Theory

    -
    - -

    -

    This leads to

    $$ \sum_{P'H'}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda C_H^{P}=0, @@ -1209,14 +1012,10 @@

    Full Configuration Interaction Th $$

    leading to the identification \( \lambda = E \).

    -

    -









    Full Configuration Interaction Theory

    -
    - -

    +

    An alternative way to derive the last equation is to start from

    $$ (\hat{H} -E)|\Psi_0\rangle = (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0, @@ -1228,14 +1027,10 @@

    Full Configuration Interaction Th of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator \( \hat{C} \).

    -

    -









    FCI and the exponential growth

    -
    - -

    +

    Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.

    The total number of Slater determinants which can be built with say \( N \) neutrons distributed among \( n \) single particle states is

    @@ -1249,8 +1044,6 @@

    FCI and the exponential growth

    $$

    and multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality \( d \) of \( d\sim 10^{18} \).

    -
    -









    Exponential wall

    @@ -1268,9 +1061,7 @@

    Exponential wall

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1283,7 +1074,10 @@

    A non-practical w (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the + +









    +

    Using the Condon-Slater rule

    +

    If we assume that we have a two-body operator at most, using the Condon-Slater rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ @@ -1300,14 +1094,10 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1319,23 +1109,24 @@

    A non-practical w $$ (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ \langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+ -\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0, +\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0. $$ -

    or

    + +









    +

    Slight rewrite

    + +

    Which we can rewrite

    $$ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}, @@ -1344,16 +1135,14 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -









    Rewriting the FCI equation

    -

    In our notes on Hartree-Fock calculations, -we have already computed the matrix \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then the matrix elements +

    In our discussions of the Hartree-Fock method planned for week 39, +we are going to compute the elements \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then these quantities result in \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and we are left with a correlation energy given by

    $$ @@ -1379,27 +1168,27 @@

    Rewriting the FCI equation











    Rewriting the FCI equation, does not stop here

    -
    - -

    +

    We need more equations. Our next step is to set up

    $$ \langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ -\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0, +\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0. $$ -

    as this equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    + +









    +

    Finding the coefficients

    +

    This equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    $$ \langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ \sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a. $$ -










    -

    Rewriting the FCI equation, please stop here

    +

    Rewriting the FCI equation

    @@ -1472,18 +1261,18 @@

    Summarizing FCI a









    Definition of the correlation energy

    -
    - -

    +

    The correlation energy is defined as, with a two-body Hamiltonian,

    $$ \Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+ \sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}. $$ -

    The coefficients \( C \) result from the solution of the eigenvalue problem. -The energy of say the ground state is then -

    +

    The coefficients \( C \) result from the solution of the eigenvalue problem.

    + +









    +

    Ground state energy

    +

    The energy of say the ground state is then

    $$ E=E_{ref}+\Delta E, $$ @@ -1492,7 +1281,6 @@

    Definition of the correlation ener $$ E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle. $$ -

    diff --git a/doc/pub/week38/html/week38.html b/doc/pub/week38/html/week38.html index 8b0befeb..c384ff65 100644 --- a/doc/pub/week38/html/week38.html +++ b/doc/pub/week38/html/week38.html @@ -211,54 +211,35 @@ 2, None, 'creation-operators-in-terms-of-pauli-matrices'), - ('Full configuration interaction theory', + ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Slater determinants as basis states, Repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states, repetition', - 2, - None, - 'slater-determinants-as-basis-states-repetition'), - ('Slater determinants as basis states', - 2, - None, - 'slater-determinants-as-basis-states'), - ('Slater determinants as basis states', + ('One-particle-one-hole state', 2, None, - 'slater-determinants-as-basis-states'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), - ('Quick repetition of the occupation representation', - 2, - None, - 'quick-repetition-of-the-occupation-representation'), + 'one-particle-one-hole-state'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), - ('Full Configuration Interaction Theory', + ('Intermediate normalization', 2, None, - 'full-configuration-interaction-theory'), + 'intermediate-normalization'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Compact expression of correlated part', + 2, + None, + 'compact-expression-of-correlated-part'), ('Full Configuration Interaction Theory', 2, None, 'full-configuration-interaction-theory'), + ('Minimization', 2, None, 'minimization'), ('Full Configuration Interaction Theory', 2, None, @@ -276,6 +257,10 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Using the Condon-Slater rule', + 2, + None, + 'using-the-condon-slater-rule'), ('A non-practical way of solving the eigenvalue problem', 2, None, @@ -284,6 +269,7 @@ 2, None, 'a-non-practical-way-of-solving-the-eigenvalue-problem'), + ('Slight rewrite', 2, None, 'slight-rewrite'), ('Rewriting the FCI equation', 2, None, @@ -296,10 +282,11 @@ 2, None, 'rewriting-the-fci-equation-does-not-stop-here'), - ('Rewriting the FCI equation, please stop here', + ('Finding the coefficients', 2, None, 'finding-the-coefficients'), + ('Rewriting the FCI equation', 2, None, - 'rewriting-the-fci-equation-please-stop-here'), + 'rewriting-the-fci-equation'), ('Rewriting the FCI equation, more to add', 2, None, @@ -315,7 +302,8 @@ ('Definition of the correlation energy', 2, None, - 'definition-of-the-correlation-energy')]} + 'definition-of-the-correlation-energy'), + ('Ground state energy', 2, None, 'ground-state-energy')]} end of tocinfo --> @@ -963,199 +951,19 @@

    Creation operators in ter











    -

    Full configuration interaction theory

    - -

    We start with a reminder on determinants in the number representation.

    - -









    -

    Slater determinants as basis states, Repetition

    -
    - -

    -

    The simplest possible choice for many-body wavefunctions are product wavefunctions. -That is -

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots -$$ - -

    because we are really only good at thinking about one particle at a time. Such -product wavefunctions, without correlations, are easy to -work with; for example, if the single-particle states \( \phi_i(x) \) are orthonormal, then -the product wavefunctions are easy to orthonormalize. -

    - -

    Similarly, computing matrix elements of operators are relatively easy, because the -integrals factorize. -

    - -

    The price we pay is the lack of correlations, which we must build up by using many, many product -wavefunctions. (Thus we have a trade-off: compact representation of correlations but -difficult integrals versus easy integrals but many states required.) -

    -
    - - -









    -

    Slater determinants as basis states, repetition

    -
    - -

    -

    Because we have fermions, we are required to have antisymmetric wavefunctions, e.g.

    -$$ -\Psi(x_1, x_2, x_3, \ldots, x_N) = - \Psi(x_2, x_1, x_3, \ldots, x_N) -$$ - -

    etc. This is accomplished formally by using the determinantal formalism

    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Product wavefunction + antisymmetry = Slater determinant.

    -
    - - -









    -

    Slater determinants as basis states

    -
    - -

    -$$ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -$$ - -

    Properties of the determinant (interchange of any two rows or -any two columns yields a change in sign; thus no two rows and no -two columns can be the same) lead to the Pauli principle: -

    - -
      -
    • No two particles can be at the same place (two columns the same); and
      • -
    • No two particles can be in the same state (two rows the same).
      • -
    -
    - - -









    -

    Slater determinants as basis states

    -
    - -

    -

    As a practical matter, however, Slater determinants beyond \( N=4 \) quickly become -unwieldy. Thus we turn to the occupation representation or second quantization to simplify calculations. -

    - -

    The occupation representation or number representation, using fermion creation and annihilation -operators, is compact and efficient. It is also abstract and, at first encounter, not easy to -internalize. It is inspired by other operator formalism, such as the ladder operators for -the harmonic oscillator or for angular momentum, but unlike those cases, the operators do not have coordinate space representations. -

    - -

    Instead, one can think of fermion creation/annihilation operators as a game of symbols that -compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater -determinants. -

    -
    - - -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). -(Note: this requirement, and others, can be relaxed, but leads to a -more involved formalism.) Any orthonormal set will do. -

    - -

    To each single-particle state \( \phi_i(x) \) we associate a creation operator -\( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \). -

    - -

    When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes -a particle to occupy the single-particle state \( \phi_i(x) \): -

    -$$ -\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle -$$ -
    - - -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    But with multiple creation operators we can occupy multiple states:

    -$$ -\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime}) -\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle. -$$ - -

    Now we impose antisymmetry, by having the fermion operators satisfy anticommutation relations:

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i -= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+ -= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0 -$$ - -

    so that

    -$$ -\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i -$$ -
    - +

    Full Configuration Interaction Theory

    -









    -

    Quick repetition of the occupation representation

    -
    - -

    -

    Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), -enforcing the Pauli exclusion principle. Thus when writing a Slater determinant -using creation operators, -

    +

    We have defined the ansatz for the ground state as

    $$ -\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle +|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, $$ -

    each index \( i,j,k, \ldots \) must be unique.

    - -

    For some relevant exercises with solutions see chapter 8 of Lecture Notes in Physics, volume 936.

    -
    - +

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions.











    -

    Full Configuration Interaction Theory

    -
    - -

    -

    We have defined the ansatz for the ground state as

    -$$ -|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle, -$$ +

    One-particle-one-hole state

    -

    where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions. -A given one-particle-one-hole (\( 1p1h \)) state can be written as -

    +

    A given one-particle-one-hole (\( 1p1h \)) state can be written as

    $$ |\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle, $$ @@ -1169,14 +977,11 @@

    Full Configuration Interaction Th $$ |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    We can then expand our exact state function for the ground state as

    @@ -1190,6 +995,9 @@

    Full Configuration Interaction Th \hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots $$ + +









    +

    Intermediate normalization

    Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have

    @@ -1201,14 +1009,11 @@

    Full Configuration Interaction Th $$ |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    We rewrite

    $$ |\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots, @@ -1219,9 +1024,12 @@

    Full Configuration Interaction Th |\Psi_0\rangle=\sum_{PH}C_H^P\Phi_H^P=\left(\sum_{PH}C_H^P\hat{A}_H^P\right)|\Phi_0\rangle, $$ -

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states. -We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators. -

    +

    where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states.

    + +









    +

    Compact expression of correlated part

    + +

    We have introduced the operator \( \hat{A}_H^P \) which contains an equal number of creation and annihilation operators.

    Our requirement of unit normalization gives

    $$ @@ -1232,14 +1040,11 @@

    Full Configuration Interaction Th $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. $$ -











    Full Configuration Interaction Theory

    -
    - -

    +

    Normally

    $$ E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}, @@ -1252,9 +1057,12 @@

    Full Configuration Interaction Th \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle, $$ -

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system. -The minimization process results in -

    +

    where \( \lambda \) is a variational multiplier to be identified with the energy of the system.

    + +









    +

    Minimization

    + +

    The minimization process results in

    $$ \delta\left[ \langle \Psi_0 | \hat{H} |\Psi_0 \rangle-\lambda \langle \Psi_0 |\Psi_0 \rangle\right]=0, $$ @@ -1264,15 +1072,10 @@

    Full Configuration Interaction Th \sum_{P'H'}\left\{\delta[C_H^{*P}]\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}- \lambda( \delta[C_H^{*P}]C_{H'}^{P'}]\right\} = 0. $$ -











    Full Configuration Interaction Theory

    -
    - -

    -

    This leads to

    $$ \sum_{P'H'}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda C_H^{P}=0, @@ -1286,14 +1089,10 @@

    Full Configuration Interaction Th $$

    leading to the identification \( \lambda = E \).

    -

    -









    Full Configuration Interaction Theory

    -
    - -

    +

    An alternative way to derive the last equation is to start from

    $$ (\hat{H} -E)|\Psi_0\rangle = (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0, @@ -1305,14 +1104,10 @@

    Full Configuration Interaction Th of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator \( \hat{C} \).

    -

    -









    FCI and the exponential growth

    -
    - -

    +

    Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.

    The total number of Slater determinants which can be built with say \( N \) neutrons distributed among \( n \) single particle states is

    @@ -1326,8 +1121,6 @@

    FCI and the exponential growth

    $$

    and multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality \( d \) of \( d\sim 10^{18} \).

    -
    -









    Exponential wall

    @@ -1345,9 +1138,7 @@

    Exponential wall

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1360,7 +1151,10 @@

    A non-practical w (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the + +









    +

    Using the Condon-Slater rule

    +

    If we assume that we have a two-body operator at most, using the Condon-Slater rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ @@ -1377,14 +1171,10 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    To see this, we look at the contributions arising from

    $$ \langle \Phi_H^P | = \langle \Phi_0|, @@ -1396,23 +1186,24 @@

    A non-practical w $$ (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. $$ -

    A non-practical way of solving the eigenvalue problem

    -
    - -

    +

    If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then

    $$ \langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+ -\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0, +\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0. $$ -

    or

    + +









    +

    Slight rewrite

    + +

    Which we can rewrite

    $$ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}, @@ -1421,16 +1212,14 @@

    A non-practical w

    where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.

    -

    -









    Rewriting the FCI equation

    -

    In our notes on Hartree-Fock calculations, -we have already computed the matrix \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then the matrix elements +

    In our discussions of the Hartree-Fock method planned for week 39, +we are going to compute the elements \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then these quantities result in \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and we are left with a correlation energy given by

    $$ @@ -1456,27 +1245,27 @@

    Rewriting the FCI equation











    Rewriting the FCI equation, does not stop here

    -
    - -

    +

    We need more equations. Our next step is to set up

    $$ \langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ -\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0, +\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0. $$ -

    as this equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    + +









    +

    Finding the coefficients

    +

    This equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as

    $$ \langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ \sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a. $$ -










    -

    Rewriting the FCI equation, please stop here

    +

    Rewriting the FCI equation

    @@ -1549,18 +1338,18 @@

    Summarizing FCI a









    Definition of the correlation energy

    -
    - -

    +

    The correlation energy is defined as, with a two-body Hamiltonian,

    $$ \Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+ \sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}. $$ -

    The coefficients \( C \) result from the solution of the eigenvalue problem. -The energy of say the ground state is then -

    +

    The coefficients \( C \) result from the solution of the eigenvalue problem.

    + +









    +

    Ground state energy

    +

    The energy of say the ground state is then

    $$ E=E_{ref}+\Delta E, $$ @@ -1569,7 +1358,6 @@

    Definition of the correlation ener $$ E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle. $$ -

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zb&c2sXhj-G6y>XP=H=Oem_1Hp?LTau5$D-)-;S8_w#T<<)sTta?8K*HJARd?Y*ERq z8kMq_fW(u7njOC4ASbYSyd-|1v1A`C2+4CNo*y8!32k%oHkl1h2HJd}!C$>V4n0(CT0IGBm#h+n z)~VPuPrpw48{GNVKJ^_!w%pI2tO|X1mXa$#gJCiv9@AtXjjHTJHXYkk=z82IgZILD z{wh^C(|lwLmGd__Pt_I~hY2rb+HHsIfSQ=xo{D0N2m(88|LR-G7ly8L5)i$L{;rbm zu7msT20hM&C#`6?ywL;e)z#4;y-4)L`{-$Qx^Cf=2S1D5wF{gj6FDPi`IlZUCtg~k zAd87J)w?85aqdJLYM!CC{GRMv92CYWB#mf=h{!?}Hn?a24kP_>XJK49@0SnPJzD#q zie_OrbMm6KjQy28z#`aolE@xK+|6_F-`@7!K%3cGIUN}PF`MAIyxEE HkEs6#XM6=- diff --git a/doc/src/week38/week38.do.txt b/doc/src/week38/week38.do.txt index 1b6337f2..6320016d 100644 --- a/doc/src/week38/week38.do.txt +++ b/doc/src/week38/week38.do.txt @@ -596,173 +596,10 @@ identity matrix on all qubits other than $i$ and $j$, ${\bf 1}\otimes being the identity matrices of appropriate dimension. -!split -===== Full configuration interaction theory ===== - -We start with a reminder on determinants in the number representation. - -!split -===== Slater determinants as basis states, Repetition ===== -!bblock -The simplest possible choice for many-body wavefunctions are _product_ wavefunctions. -That is -!bt -\[ -\Psi(x_1, x_2, x_3, \ldots, x_N) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots -\] -!et -because we are really only good at thinking about one particle at a time. Such -product wavefunctions, without correlations, are easy to -work with; for example, if the single-particle states $\phi_i(x)$ are orthonormal, then -the product wavefunctions are easy to orthonormalize. - -Similarly, computing matrix elements of operators are relatively easy, because the -integrals factorize. - - -The price we pay is the lack of correlations, which we must build up by using many, many product -wavefunctions. (Thus we have a trade-off: compact representation of correlations but -difficult integrals versus easy integrals but many states required.) -!eblock - -!split -===== Slater determinants as basis states, repetition ===== -!bblock -Because we have fermions, we are required to have antisymmetric wavefunctions, e.g. -!bt -\[ -\Psi(x_1, x_2, x_3, \ldots, x_N) = - \Psi(x_2, x_1, x_3, \ldots, x_N) -\] -!et -etc. This is accomplished formally by using the determinantal formalism -!bt -\[ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -\] -!et -Product wavefunction + antisymmetry = Slater determinant. -!eblock - -!split -===== Slater determinants as basis states ===== -!bblock -!bt -\[ -\Psi(x_1, x_2, \ldots, x_N) -= \frac{1}{\sqrt{N!}} -\det \left | -\begin{array}{cccc} -\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_N) \\ -\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_N) \\ - \vdots & & & \\ -\phi_N(x_1) & \phi_N(x_2) & \ldots & \phi_N(x_N) -\end{array} -\right | -\] -!et -Properties of the determinant (interchange of any two rows or -any two columns yields a change in sign; thus no two rows and no -two columns can be the same) lead to the Pauli principle: - -* No two particles can be at the same place (two columns the same); and -* No two particles can be in the same state (two rows the same). - -!eblock - - -!split -===== Slater determinants as basis states ===== -!bblock -As a practical matter, however, Slater determinants beyond $N=4$ quickly become -unwieldy. Thus we turn to the _occupation representation_ or _second quantization_ to simplify calculations. - -The occupation representation or number representation, using fermion _creation_ and _annihilation_ -operators, is compact and efficient. It is also abstract and, at first encounter, not easy to -internalize. It is inspired by other operator formalism, such as the ladder operators for -the harmonic oscillator or for angular momentum, but unlike those cases, the operators _do not have coordinate space representations_. - -Instead, one can think of fermion creation/annihilation operators as a game of symbols that -compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater -determinants. -!eblock - -!split -===== Quick repetition of the occupation representation ===== -!bblock -We start with a set of orthonormal single-particle states $\{ \phi_i(x) \}$. -(Note: this requirement, and others, can be relaxed, but leads to a -more involved formalism.) _Any_ orthonormal set will do. - -To each single-particle state $\phi_i(x)$ we associate a creation operator -$\hat{a}^\dagger_i$ and an annihilation operator $\hat{a}_i$. - -When acting on the vacuum state $| 0 \rangle$, the creation operator $\hat{a}^\dagger_i$ causes -a particle to occupy the single-particle state $\phi_i(x)$: -!bt -\[ -\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle -\] -!et -!eblock - -!split -===== Quick repetition of the occupation representation ===== -!bblock -But with multiple creation operators we can occupy multiple states: -!bt -\[ -\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime}) -\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle. -\] -!et - -Now we impose antisymmetry, by having the fermion operators satisfy _anticommutation relations_: -!bt -\[ -\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i -= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+ -= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0 -\] -!et -so that -!bt -\[ -\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i -\] -!et -!eblock - - -!split -===== Quick repetition of the occupation representation ===== -!bblock -Because of this property, automatically $\hat{a}^\dagger_i \hat{a}^\dagger_i = 0$, -enforcing the Pauli exclusion principle. Thus when writing a Slater determinant -using creation operators, -!bt -\[ -\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle -\] -!et -each index $i,j,k, \ldots$ must be unique. - -For some relevant exercises with solutions see chapter 8 of "Lecture Notes in Physics, volume 936":"http://www.springer.com/us/book/9783319533353". - -!eblock - !split ===== Full Configuration Interaction Theory ===== -!bblock + We have defined the ansatz for the ground state as !bt \[ @@ -770,6 +607,10 @@ We have defined the ansatz for the ground state as \] !et where the index $i$ defines different single-particle states up to the Fermi level. We have assumed that we have $N$ fermions. + +!split +===== One-particle-one-hole state ===== + A given one-particle-one-hole ($1p1h$) state can be written as !bt \[ @@ -788,11 +629,11 @@ and a general $NpNh$ state as |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. \] !et -!eblock + !split ===== Full Configuration Interaction Theory ===== -!bblock + We can then expand our exact state function for the ground state as !bt @@ -807,6 +648,9 @@ where we have introduced the so-called correlation operator \hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots \] !et + +!split +===== Intermediate normalization ===== Since the normalization of $\Psi_0$ is at our disposal and since $C_0$ is by hypothesis non-zero, we may arbitrarily set $C_0=1$ with corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have !bt @@ -820,11 +664,11 @@ resulting in |\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle. \] !et -!eblock + !split ===== Full Configuration Interaction Theory ===== -!bblock + We rewrite !bt \[ @@ -838,6 +682,10 @@ in a more compact form as \] !et where $H$ stands for $0,1,\dots,n$ hole states and $P$ for $0,1,\dots,n$ particle states. + +!split +===== Compact expression of correlated part ===== + We have introduced the operator $\hat{A}_H^P$ which contains an equal number of creation and annihilation operators. Our requirement of unit normalization gives @@ -852,11 +700,11 @@ and the energy can be written as E= \langle \Psi_0 | \hat{H} |\Psi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. \] !et -!eblock + !split ===== Full Configuration Interaction Theory ===== -!bblock + Normally !bt \[ @@ -872,6 +720,10 @@ is equivalent to finding the variational minimum of \] !et where $\lambda$ is a variational multiplier to be identified with the energy of the system. + +!split +===== Minimization ===== + The minimization process results in !bt \[ @@ -885,14 +737,12 @@ and since the coefficients $\delta[C_H^{*P}]$ and $\delta[C_{H'}^{P'}]$ are comp \lambda( \delta[C_H^{*P}]C_{H'}^{P'}]\right\} = 0. \] !et -!eblock + !split ===== Full Configuration Interaction Theory ===== -!bblock - This leads to !bt \[ @@ -909,13 +759,10 @@ If we then multiply by the corresponding $C_H^{*P}$ and sum over $PH$ we obtain !et leading to the identification $\lambda = E$. -!eblock - - !split ===== Full Configuration Interaction Theory ===== -!bblock + An alternative way to derive the last equation is to start from !bt \[ @@ -927,13 +774,13 @@ results. As stated previously, one solves this equation normally by diagonaliz numerically exactly) in a large Hilbert space (it will be truncated in terms of the number of single-particle states included in the definition of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator $\hat{C}$. -!eblock + !split ===== FCI and the exponential growth ===== -!bblock + Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly. The total number of Slater determinants which can be built with say $N$ neutrons distributed among $n$ single particle states is @@ -943,6 +790,7 @@ The total number of Slater determinants which can be built with say $N$ neutrons \] !et + For a model space which comprises the first for major shells only $0s$, $0p$, $1s0d$ and $1p0f$ we have $40$ single particle states for neutrons and protons. For the eight neutrons of oxygen-16 we would then have !bt \[ @@ -950,7 +798,7 @@ For a model space which comprises the first for major shells only $0s$, $0p$, $1 \] !et and multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality $d$ of $d\sim 10^{18}$. -!eblock + !split @@ -965,7 +813,7 @@ This number can be reduced if we look at specific symmetries only. However, the !split ===== A non-practical way of solving the eigenvalue problem ===== -!bblock + To see this, we look at the contributions arising from !bt \[ @@ -979,7 +827,10 @@ from the left in (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. \] !et -If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the + +!split +===== Using the Condon-Slater rule ===== +If we assume that we have a two-body operator at most, using the Condon-Slater rule gives then an equation for the correlation energy in terms of $C_i^a$ and $C_{ij}^{ab}$ only. We get then !bt \[ @@ -996,12 +847,12 @@ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a} !et where the energy $E_0$ is the reference energy and $\Delta E$ defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian. -!eblock + !split ===== A non-practical way of solving the eigenvalue problem ===== -!bblock + To see this, we look at the contributions arising from !bt \[ @@ -1015,21 +866,25 @@ from the left in (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0. \] !et -!eblock + !split ===== A non-practical way of solving the eigenvalue problem ===== -!bblock + If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of $C_i^a$ and $C_{ij}^{ab}$ only. We get then !bt \[ \langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+ -\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0, +\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0. \] !et -or + +!split +===== Slight rewrite ===== + +Which we can rewrite !bt \[ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ @@ -1038,15 +893,15 @@ E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a} !et where the energy $E_0$ is the reference energy and $\Delta E$ defines the so-called correlation energy. The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian. -!eblock + !split ===== Rewriting the FCI equation ===== !bblock -In our notes on Hartree-Fock calculations, -we have already computed the matrix $\langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle $ and $\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle$. If we are using a Hartree-Fock basis, then the matrix elements +In our discussions of the Hartree-Fock method planned for week 39, +we are going to compute the elements $\langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle $ and $\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle$. If we are using a Hartree-Fock basis, then these quantities result in $\langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0$ and we are left with a *correlation energy* given by !bt \[ @@ -1070,16 +925,19 @@ This equation determines the correlation energy but not the coefficients $C$. !split ===== Rewriting the FCI equation, does not stop here ===== -!bblock + We need more equations. Our next step is to set up !bt \[ \langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+ \sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+ -\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0, +\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0. \] !et -as this equation will allow us to find an expression for the coefficents $C_i^a$ since we can rewrite this equation as + +!split +===== Finding the coefficients ===== +This equation will allow us to find an expression for the coefficents $C_i^a$ since we can rewrite this equation as !bt \[ \langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+ @@ -1087,10 +945,10 @@ as this equation will allow us to find an expression for the coefficents $C_i^a$ \sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a. \] !et -!eblock + !split -===== Rewriting the FCI equation, please stop here ===== +===== Rewriting the FCI equation ===== !bblock We see that on the right-hand side we have the energy $E$. This leads to a non-linear equation in the unknown coefficients. These equations are normally solved iteratively ( that is we can start with a guess for the coefficients $C_i^a$). A common choice is to use perturbation theory for the first guess, setting thereby @@ -1155,7 +1013,7 @@ If we can diagonalize large matrices, FCI is the method of choice since: !split ===== Definition of the correlation energy ===== -!bblock + The correlation energy is defined as, with a two-body Hamiltonian, !bt \[ @@ -1164,6 +1022,9 @@ The correlation energy is defined as, with a two-body Hamiltonian, \] !et The coefficients $C$ result from the solution of the eigenvalue problem. + +!split +===== Ground state energy ===== The energy of say the ground state is then !bt \[ @@ -1177,9 +1038,6 @@ E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle. \] !et -!eblock - -