Ctrl+K

Problems

Problems#

Session 1#

1. The direct product of matrices is also known as the Kronecker product and it is defined as

\[\begin{split} \mathbf{A} \otimes \mathbf{B} = \begin{pmatrix} a_{11} \mathbf{B} & \cdots & a_{n1} \mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{n1} \mathbf{B} & \cdots & a_{nn} \mathbf{B} \\ \end{pmatrix} \end{split}\]

Two important identities are

\[\begin{eqnarray*} \mathbf{A} \otimes (\mathbf{B} + \mathbf{C} ) & = & \mathbf{A} \otimes \mathbf{B} + \mathbf{A} \otimes \mathbf{C} \\ (\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) & = & (\mathbf{A} \mathbf{C}) \otimes (\mathbf{B} \mathbf{D}) \end{eqnarray*}\]

Let \(\mathbf{A}\) and \(\mathbf{B}\) be operator matrices in charge conjugation and spin spaces of a single electron, respectively. Let \(\mathbf{C}\) and \(\mathbf{D}\) be two corresponding state vectors. Show the second identity by explicit matrix calculations.

2. Generators for rotations of electronic wave functions are the angular momentum operators. A finite rotation an angle \(\phi\) about axis \(\mathbf{n}\) is achieved by the total angular momentum operator

\[ \hat{R}(\phi, \mathbf{n} ) = \exp(-i\frac{\phi}{\hbar}(\mathbf{n} \cdot \hat{\mathbf{j}})) = \exp(-i\frac{\phi}{\hbar}(\mathbf{n} \cdot \hat{\mathbf{l}})) \otimes \exp(-i\frac{\phi}{\hbar}(\mathbf{n} \cdot \hat{\mathbf{s}})) \]

Show that

\[ \hat{\mathbf{j}} = \hat{\mathbf{l}} \otimes \hat{\mathbf{I}} + \hat{\mathbf{I}} \otimes \hat{\mathbf{s}} \]

or, in other words,

\[ e^{\mathbf{A}} \otimes e^{\mathbf{B}} = e^{\mathbf{A}\otimes \mathbf{I} + \mathbf{I} \otimes \mathbf{B}} \]

3. The rotation operator in spin space, resulting in a rotation an angle \(\phi\) about axis \(\mathbf{n}\), is

\[ R(\phi, \mathbf{n}) = e^{-i\phi(\mathbf{n} \cdot \boldsymbol{\sigma})/2} \]

where the Cartesian components of the spin operator, \(\boldsymbol{\sigma}\), are the Pauli spin matrices. Show that

\[ R(\phi, \mathbf{n}) = \cos(\phi/2) \mathbf{I} - i \sin(\phi/2) (\mathbf{n} \cdot \boldsymbol{\sigma}) \]

4. A state vector in a two-electron spin-space is given by

\[\begin{split} | \Psi \rangle = \begin{pmatrix} \alpha_1 \\ \beta_1 \end{pmatrix} \otimes \begin{pmatrix} \alpha_2 \\ \beta_2 \end{pmatrix} = \begin{pmatrix} \alpha_1 \alpha_2 \\ \alpha_1 \beta_2 \\ \beta_1 \alpha_2 \\ \beta_1 \beta_2 \end{pmatrix} \end{split}\]

(a) What are the explicit forms of the \(z\)-component of the spin operator, \(\hat{S}_z\), expressed with and without use of direct products?

(b) Is \(\hat{S}_z\) a one- or two-electron operator?

(c) Determine \(\hat{S}_z | \Psi \rangle\).

(d) Is \(| \Psi \rangle\) an eigenvector to \(\hat{S}_z\)?

(e) What are the eigenvalues and eigenvectors of \(\hat{S}_z\)?

(f) Are the eigenvectors unique?

Extra

Consider the operator

\[\begin{split} \hat{\Omega} = \begin{pmatrix} a & c\\ c & b\\ \end{pmatrix} ; \quad a,b,c>0; \quad b>c \end{split}\]

(a) Based on a starting vector, \(| {\Psi}_1 \rangle\), pointing in a predefined direction, design a scheme to define a bi-orthogonal basis. The basis should fulfill

\[\begin{eqnarray*} | \Phi_i \rangle = \hat{\Omega} | \Psi_i \rangle; \quad \langle \Psi_i | \Phi_j \rangle = \delta_{ij} \end{eqnarray*}\]

(b) In this basis, show that

\[ \hat{\Omega} = \sum_i | \Phi_i \rangle\langle \Phi_i | \]

(c) Implement your scheme in Python and demonstrate that it numerically works for

\[\begin{split} \hat{\Omega} = \begin{pmatrix} 3 & 1\\1 & 7\\ \end{pmatrix}; \quad % | \Psi_1 \rangle = N \begin{pmatrix} 1\\2\\ \end{pmatrix} \end{split}\]

where \(N\) is a normalization constant.

(d) Describe how the situation changes when the starting vector is chosen as an eigenvector of \(\hat{\Omega}\).

Session 2#

1. Let us define the one-electron density operator as

\[ \hat{n}(\mathbf{r}) = \sum_{i=1}^N \delta(\mathbf{r} - \mathbf{r}_i) \]

Show that

\[\begin{equation*} \label{eq:den1-oper} n(\mathbf{r}) = \langle \Psi | \hat{n}(\mathbf{r}) | \Psi \rangle \end{equation*}\]

is in agreement with the definition of the one-electron density based on a probabilistic interpretation of the wave function.

2. For an \(N\)-electron system in a state described by a Slater determinant, show that the one-electron density becomes

\[ n(\mathbf{r}) = \sum_{i=1}^N |\psi_i(\mathbf{r})|^2 \]

Reflect on your results in terms of a system of independent particles.

3. For an \(N\)-electron system in a state described by a Slater determinant, show that the two-electron density becomes

\[\begin{equation*} n(\mathbf{r}_1, \mathbf{r}_2) = \sum_{i=1}^N \sum_{j=1}^N \left[\rule{0pt}{12pt} |\psi_i(\mathbf{r}_1)|^2 |\psi_j(\mathbf{r}_2)|^2 - \psi_i^\dagger(\mathbf{r}_1) \psi_j^\dagger(\mathbf{r}_2) \psi_j(\mathbf{r}_1) \psi_i(\mathbf{r}_2) \right] \end{equation*}\]

Reflect on your results in terms of a system of independent particles. Specifically focus the discussion around orbital pairs \((i,j)\) with identical and opposite spins, respectively.

Session 3#

1. Consider the one-electron operator

\[ \hat{\Omega} = \sum_{i=1}^N \hat{\omega}(i) \]

Show that the transition matrix element for \(\hat{\Omega}\) between a reference Slater determinant and a corresponding single electron excited determinant equals

\[ \langle \Psi_i^s | \hat{\Omega} | \Psi \rangle = \langle \psi_s | \hat{\omega} | \psi_i \rangle \]

where orbitals \(i\) and \(s\) are occupied and unoccupied in the reference state, respectively. Reflect on your results in terms of a system of independent particles. If \(\hat{\Omega}\) is a pure orbital operator, what is the implication on the transition matrix elements?

2. Repeat the steps in the preious exercise and derive an expression for transition moments between a reference determinant and an associated doubly-excited determinant, i.e.,

\[ \langle \Psi_{ij}^{st} | \hat{\Omega} | \Psi \rangle \]

3. Derive an explicit expression for the HF energy in terms of one- and two-electron integrals. Use the following notation for the \(N\)-electron Hamiltonian

\[ \hat{H} = \sum_{i=1}^N \hat{h}(i) + \sum_{j>i}^N \hat{g}(i,j) \]

For the specific (and important) case of a closed-shell HF ground state, simplify your result so that the final expression is given in terms of integrals with respect to spatial orbitals.

Extra

Let \(|\Psi \rangle\) be a Slater determinant and \(\hat{\Omega}\) a two-electron operator

\[ \hat{\Omega} = \sum_{j>i}^N \hat{\omega}(i,j) \]

Show the following relations for the corresponding integrals

\[\begin{align*} \langle \Psi | \hat{\Omega} | \Psi \rangle &= \frac{1}{2} \sum_{i,j}^N \left[\rule{0pt}{12pt} \langle ij| \hat{\omega} | ij \rangle - \langle ij| \hat{\omega} |ji \rangle \right] \\ \langle \Psi | \hat{\Omega} | \Psi_{i}^{s} \rangle &= \sum_{j}^N \left[\rule{0pt}{12pt} \langle ij| \hat{\omega} |sj \rangle - \langle ij| \hat{\omega} |js \rangle \right] \\ \langle \Psi | \hat{\Omega} | \Psi_{ij}^{st} \rangle &= \langle ij| \hat{\omega} |st \rangle - \langle ij| \hat{\omega} |ts \rangle \end{align*}\]

Session 4#

1. The Fock operator reads

\[ \hat{f} = \hat{h} + \sum_{i=1}^N \left( \hat{J}_i - \hat{K}_i \right) \]

where

\[\begin{eqnarray*} \hat{J}_i | \psi_a \rangle & = & \left[ \int \frac{ \left|\psi_i(\mathbf{r}')\right|^2 }{ |\mathbf{r} - \mathbf{r}'| } d^3\mathbf{r}' \right] | \psi_a \rangle \\ % \hat{K}_i | \psi_a \rangle & = & \left[ \int \frac{ \psi_i^\dagger(\mathbf{r}') \psi_a(\mathbf{r}') }{ |\mathbf{r} - \mathbf{r}'| } d^3\mathbf{r}' \right] | \psi_i \rangle \end{eqnarray*}\]

Show that \(\hat{f}\) is Hermitian.

2. Introducing a Lagrangian

\[ \mathcal{L}[\{\psi_i\}] = E[\{\psi_i\}] - \sum_{i,j=1}^N \varepsilon_{ji} (\langle \psi_i|\psi_j \rangle - \delta_{ij}) \]

Show that the condition that the first variation in the Lagrangian vanishes, i.e., \(\delta \mathcal{L} = 0\), leads to the following equation from which we can determine the set of orbitals

\[ \hat{f} |\psi_i \rangle = \sum_{j=1}^N \varepsilon_{ji} |\psi_j \rangle \]

Tip: You will need the previously derived expression for the HF energy.

Session 5#

1. Introduce a unitary transformation of the occupied orbitals

\[ |\psi_i'\rangle = \sum_{j=1}^N |\psi_j\rangle U_{ji} \]

(a) Show that the Fock operator, \(\hat{f}\), is invariant under this orbital transformation.

(b) Choose a unitary transformation such that the matrix representation of the Fock operator becomes diagonal. Show that the Hartree–Fock equation in this basis takes the form

\[ \hat{f} |\psi_a \rangle = \varepsilon_{a} |\psi_a \rangle \]

2. Show that ionization potentials and electron affinities are given by minus the orbital energies, i.e.,

\[\begin{align*} \mathrm{IP} = E^{N-1}_i - E_\mathrm{HF}^N &= -\varepsilon_i \\ \mathrm{EA} = E_\mathrm{HF}^N - E^{N+1}_s &= -\varepsilon_s \end{align*}\]

This result is known as Koopmans’ theorem.

Session 6#

1. Introduce a set of non-orthogonal basis functions (or atomic orbitals) denoted as \(\{\chi_\alpha\}\). In this basis, show that the matrix representation of the Fock operator can be written on the form

\[ F_{\alpha\beta} = \langle \chi_\alpha | \hat{f} | \chi_\beta \rangle = h_{\alpha\beta} + \sum_{\gamma\delta} D_{\gamma\delta} \left[\rule{0pt}{12pt} \langle \chi_\alpha \chi_\gamma | \hat{g} | \chi_\beta \chi_\delta \rangle - \langle \chi_\alpha \chi_\gamma | \hat{g} | \chi_\delta \chi_\beta \rangle \right] \]

2. Show the following relation between the one-electron density and the density matrix

\[ n(\mathbf{r}) = \sum_{\gamma\delta} D_{\gamma\delta} \chi_\gamma^*(\mathbf{r})\chi_\delta(\mathbf{r}) \]

3. Consider the Hartree–Fock equation

\[\begin{equation*} \mathbf{F}\mathbf{C} = \mathbf{S} \mathbf{C} \boldsymbol{\varepsilon} \end{equation*}\]

Introduce a non-unitary change of the AO basis

\[\begin{equation*} \chi_\alpha'(\mathbf{r}) = \sum_\beta X_{\beta\alpha} \chi_\beta(\mathbf{r}) ; \quad \mathbf{X} = \mathbf{S}^{-1/2} \end{equation*}\]

(a) Show that \(\{\chi_\alpha'\}\) forms an orthonormal set.

(b) Show that \(\mathbf{S}\) is Hermitian and positive definite.

(c) Show that the form of the Hartree–Fock equation in this new basis becomes

\[\begin{equation*} \mathbf{F}'\mathbf{C}' = \mathbf{C}' \boldsymbol{\varepsilon} \end{equation*}\]

Session 7#

1. Return to Problem 4 of Session 1.

(a) What is the explicit matrix form of the spin operator, \(\hat{S}^2\)?

(b) Is \(\hat{S}^2\) a one- or two-electron operator? How can you tell solely from the matrix representation?

(c) What are the eigenvalues and eigenvectors of \(\hat{S}^2\)?

(d) Determine the pairs of spin quantum numbers, \(S\) and \(M_S\), for the eigenvectors. Is this set of quantum numbers sufficient to uniquely label the states?

(e) Classify the states as singlets, doublets, triplets, etc.?

2. Consider an atomic two-electron system such as helium in a state described by a Slater determinant

\[\begin{equation*} | \Psi \rangle = | \psi_1\psi_2 \rangle \end{equation*}\]

where

\[\begin{equation*} \psi_1(\mathbf{r}) = R_1(r) Y_{00}(\theta,\varphi) \otimes \begin{pmatrix} 1\\0 \end{pmatrix} ; \quad \psi_2(\mathbf{r}) = R_2(r) Y_{11}(\theta,\varphi) \otimes \begin{pmatrix} 1\\0 \end{pmatrix} \end{equation*}\]

In concern with the angular momenta of this state, we can henceforth disregard the radial parts of the orbitals and only consider the spherical harmonics, \(Y_{lm}\), and the spin components. The operator of total angular momentum in \(N\)-particle space is

\[\begin{equation*} \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} \end{equation*}\]

where

\[\begin{equation*} \hat{\mathbf{J}} = \sum_{i=1}^N \hat{\mathbf{j}}(i) \end{equation*}\]

It is also convenient to introduce ladder operators that take the form

\[\begin{equation*} \hat{J}_\pm = \hat{J}_x \pm i \hat{J}_y \end{equation*}\]

(a) Show that \(\Psi\) is anti-symmetric and symmetric with respect to particle interchange in orbital and spin spaces, respectively.

(b) Show that

\[ \hat{J}^2 = \hat{J}^2_z + \hbar \hat{J}_z + \hat{J}_- \hat{J}_+ \]

(c) Show that

\[ \hat{J}_z |\Psi\rangle = 2 \hbar | \Psi \rangle \]

(d) Based on the relation

\[ \hat{J}^2 |\Psi\rangle = J(J+1) \hbar^2 | \Psi \rangle \]

Determine \(J\) as well as the corresponding values for \(L\) and \(S\).

(e) Write down the term symbol for this given state. The term symbol notation in atomic physics is defined by

\[ {}^{2S+1} L_{J} \]

where \(2S+1\) refers to the spin multiplicity and \(L\) is replaced by a letter \(\{S, P, D, F, \ldots\}\) depending on its value. This nomenclature is derived from the characteristics of the spectroscopic lines corresponding to (s, p, d, f) orbitals: sharp, principal, diffuse, and fundamental.

Extra

Consider the hydrogen molecule. In a minimal basis consisting of two \(1s\) atomic orbitals (AOs), the solution of the Hartree–Fock problem will give us two spatial molecular orbitals (MOs): the bonding and symmetric \(\phi_S\) as well as the anti-bonding and anti-symmetric \(\phi_A\). The explicit form of these orthogonal MOs will be

\[\begin{eqnarray*} \phi_S(\mathbf{r}) & = & N_S \left[\rule{0pt}{12pt} \chi_{1s}(\mathbf{r} - \mathbf{R}_1) + \chi_{1s}(\mathbf{r} - \mathbf{R}_2) \right] \\ \phi_A(\mathbf{r}) & = & N_A \left[\rule{0pt}{12pt} \chi_{1s}(\mathbf{r} - \mathbf{R}_1) - \chi_{1s}(\mathbf{r} - \mathbf{R}_2) \right] \end{eqnarray*}\]

where \(N_S\) and \(N_A\) are normalization constants.

(a) Write down all possible Slater determinants that we can form in the minimal basis. Use a shorthand notation for determinants in terms of \(|\psi_S\psi_{\bar{A}}\rangle\) to denote occupation of spin orbitals \(\psi_S(\mathbf{r}) = \phi_S(\mathbf{r})\alpha\) and \(\psi_{\bar{A}}(\mathbf{r}) = \phi_A(\mathbf{r})\beta\).

(b) Show that all determinants are eigenstates of the \(z\)-component of the many-electron spin operator, \(\hat{S}_z\). What are the eigenvalues \(M_S\), i.e., the spin projection of the molecule along the \(z\)-axis?

(c) Show that some, but not all, determinants are eigenfunctions of the magnitude of the spin operator, \(\hat{S}^2\). What are the eigenvalues of the eigenstates? Identify \(S\) from the eigenvalues \(S(S + 1)\hbar^2\).

(d) For the set of determinants that are not eigenstates of \(\hat{S}^2\), construct the spin-adapted configurations and show explicitly that these are eigenstates. Determine the eigenvalues, or rather the values of \(S\).

(e) Determine the electronic energies of all the eigenstates in the minimal basis. Express your result in terms of spatial one- and two-electron integrals. Order the states in an energy level diagram and indicate their respective spin state.

(f) Let us consider the three-fold degenerate triplet state \(|T_1 , M_S \rangle\) in the band gap. Determine the action of the ladder operators on \(|T_1, M_S = -1 \rangle\), i.e., compute \(\hat{S}_-|T_1, -1 \rangle\) and \(\hat{S}_+ | T_1, -1 \rangle\). Compare your result with the action of \(\hat{j}_\pm\) on the single-particle state that you studied in your course on Quantum Mechanics.

Session 8#

1. Consider the ethylene molecule.

(a) Use the flowchart in the Appendix to determine the molecular point group.

(b) Introduce a coordinate system with the \(x\)-axis along the carbon–carbon bond and \(z\) being the out-of-plane axis. Identify all symmetry operations of the group (ignore improper rotations).

(c) Construct the group table.

(d) Introduce a less than minimal basis of atomic orbitals by only considering the four hydrogen \(1s\)-atomic orbitals, \(\{\chi_\alpha\}^4_{\alpha=1}\). This is not an adequate basis set but serves here to illustrate the principles of symmetry. Determine the action of the symmetry operations on each and every \(\chi_\alpha\), in other words obtain the matrices \(\Gamma(\hat{G})\) determined by

\[ \hat{G}\chi_\alpha = \sum_\beta \chi_\beta \Gamma_{\beta\alpha}(\hat{G}) \]

where \(\hat{G}\) denotes a given symmetry operation. These matrices form a homomorphic, reducible, representation of the group; convince yourself why this is so.

(e) Find a unitary transformation matrix \(U\) such that the transformed matrices

\[ \Gamma^{'}(\hat{G}) = U^\dagger \Gamma(\hat{G}) U \]

become block diagonal for all \(\hat{G}\), and with blocks of dimensions as small as possible. Each of the sets of sub-matrices \(\Gamma^{(i)}(\hat{G})\) forms an irreducible representation (irrep) of the group.

(f) Find the characters of the matrices \(\Gamma^{(i)}(\hat{G})\) and set up the character table of the group. The chemical notation for one-dimensional irreps (a one-dimensional matrix is a number) is \(A\) (if the characters for all \(C_2\) operations are equal to 1) or \(B\) (otherwise). Furthermore, if inversion \(\hat{i}\) is a symmetry operation then an additional label is used, either \(g\) (gerade) or \(u\) (ungerade). Thus possible irreps may be for instance \(A_{1g}\), \(B_{2u}\), etc.

(g) By applying \(U\) to the set of basis functions \(\{\chi_\alpha\}\), construct the symmetry-adapted basis functions \(\{\chi_\alpha^\mathrm{SA}\}\).

(h) Each symmetry-adapted basis function transforms under the symmetry operators according to the characters in the character table, i.e.,

\[ \hat{G} \chi_\alpha^\mathrm{SAO} = \pm \chi_\alpha^\mathrm{SAO} \]

Compare with the rows in character table and determine which irreps are spanned by your set of symmetry-adapted basis functions.

(i) What additional AOs would you need to add to your calculation in order to span all irreps?

Session 9#

1. Consider a trans-butadiene molecule in the \(xy\)-plane.

(a) Use the flowchart in the Appendix to determine the molecular point group.

(b) Introduce a less than minimal basis of atomic orbitals by only considering the four carbon \(2p_z\)-atomic orbitals, \(\{\chi_\alpha\}^4_{\alpha=1} = \{p^\alpha_z\}^4_{\alpha=1}\). Based on experience and intuition, form the symmetry-adapted basis functions \(\{\chi_\alpha^\mathrm{SAO}\}\) and order them in energy.

(c) With the character table in the Appendix, determine to which irreducible representations the SAOs belong.

(d) Run a Jupyter notebook SCF optimization of the HF wave function

C         -1.83140       -0.13080        0.00000
C         -0.60240        0.39750        0.00000
C          0.60240       -0.39750        0.00000
C          1.83140        0.13080        0.00000
H         -2.70360        0.51430        0.00000
H         -1.99690       -1.20300        0.00000
H         -0.49790        1.47890        0.00000
H          0.49790       -1.47920        0.00000
H          1.99690        1.20300        0.00000
H          2.70360       -0.51430        0.00000

Identify the \(\pi\)- and \(\pi^*\)-orbitals in the program output and compare the printed MO coefficients with your SAOs.

(e) What are the symmetries of the HF ground state and the lowest \(\pi\pi^*\)-excited state?

(f) Which Cartesian components of the electric-dipole operator couple the two states, or in other words have

\[ \langle \Psi_\pi^{\pi^*} | \hat{\mu}_\alpha | \Psi_\mathrm{HF} \rangle \neq 0 ; \quad \alpha \in \{x,y,z\} \]

Note: This consideration is the underlying principle for spectroscopic selection rules in physics.

Session 10#

1. Consider the hydrogen molecule at the equilibrium bond length of 0.741 {\AA}. The HF ground state and the doubly excited determinant are denoted

\[ | \Psi_\mathrm{HF} \rangle = |\sigma_g, \overline{\sigma}_g \rangle = |1, \bar{1} \rangle; \quad | \Psi_{g\bar{g}}^{u\bar{u}} \rangle = |\sigma_u, \overline{\sigma}_u \rangle = |2, \bar{2} \rangle \]

In a minimal STO-3G basis set, the following integrals (in a.u.) are available in the spatial MO basis:

Integral

Value

\(h_{11}\)

\(-1.2527\)

\(h_{22}\)

\(-0.4757\)

\((11 \mid 11)\)

\(0.6746\)

\((11\mid 22)\)

\(0.6635\)

\((22\mid 22)\)

\(0.6975\)

\((12\mid 12)\)

\(0.1813\)

(a) Form the configuration interaction doubles (CID) Hamiltonian.

(b) Express the matrix elements of the CID Hamiltonian in terms of one- and two-electron integrals.

(c) Insert numerical values and determine the CID ground-state energy.

(d) Determine the CID estimate of the correlation energy both in absolute terms and relative to the HF energy.

(e) What are the percentage weights of the determinants \(| \Psi_\mathrm{HF} \rangle\) and \(| \Psi_{g\bar{g}}^{u\bar{u}} \rangle\) in the CID ground-state wave function?

Session 11#

1. Consider the setup made in Problem 1 of Session 10. Write the CID ground-state wave function on the following form

\[ |\Psi \rangle = \cos\theta | \Psi_\mathrm{HF} \rangle + \sin\theta | \Psi_{g\bar{g}}^{u\bar{u}} \rangle \]

(a) Show that

\[ E(\theta) = \cos^2\theta E_\mathrm{HF} + \sin^2\theta E_\mathrm{u\bar{u}} + \sin 2\theta \, (12|12) \]

(b) Show that the energy is minimized by

\[ \theta = \frac{1}{2} \mathrm{arctan} \left[ \frac{ 2(12|12) }{ E_\mathrm{HF} - E_\mathrm{u\bar{u}} } \right] \]

(c) Compare the minimized energy to that obtained in Problem 1 in Session 10.

2. Consider the setup made in Problem 1 of Session 10 and partition the Hamiltonian according to

\[ \hat{H} = \hat{H}_0 + \hat{V} + V^\mathrm{n,rep} \]

with

\[\begin{split} \hat{H}_0 = \begin{bmatrix} 2h_{11} & 0 \\ 0 & 2h_{22} \\ \end{bmatrix}; \quad \hat{V} = \begin{bmatrix} (11|11) & (12|12) \\ (12|12) & (22|22) \\ \end{bmatrix} \end{split}\]

The basis for Rayleigh–Schrödinger perturbation theory (RSPT) are given by the solutions to

\[ \hat{H}_0 | \Phi_n \rangle = \mathcal{E}_n | \Phi_n \rangle \]

The exact ground state and energy are expanded in orders of \(\hat{V}\):

\[ |\Psi \rangle = \sum_{k=0}^\infty |\Psi^{(n)} \rangle; \quad E = \sum_{k=0}^\infty E^{(n)} \]

(a) Determine \(\Psi^{(0)}\) and \(E^{(0)}\).

(b) Determine \(E^{(1)}\) and compare your result with the HF energy.

(c) Determine \(\Psi^{(1)}\) and \(E^{(2)}\).

(d) What are the percentage weights of the determinants \(| \Psi_\mathrm{HF} \rangle\) and \(| \Psi_{g\bar{g}}^{u\bar{u}} \rangle\) in the ground-state wave function that are correct to first order in RSPT?

Session 12#

1. Consider the setup made in Problem 1 of Session 10 and partition the Hamiltonian according to

\[ \hat{H} = \hat{F} + \hat{V} + V^\mathrm{n,rep} \]

with

\[ \hat{V} = \hat{H} - \hat{F} - V^\mathrm{n,rep} \]

The basis for Møller–Plesset perturbation theory (MPPT) are given by the solutions to

\[ \hat{F} | \Phi_n \rangle = \mathcal{E}_n | \Phi_n \rangle \]

The exact ground state and energy are expanded in orders of \(\hat{V}\):

\[ |\Psi \rangle = \sum_{k=0}^\infty |\Psi^{(n)} \rangle; \quad E = \sum_{k=0}^\infty E^{(n)}; \quad \]

(a) Determine \(\Psi^{(0)}\) and \(E^{(0)}\).

(b) Determine \(E^{(1)}\) and compare your result with the HF energy.

(c) Determine \(\Psi^{(1)}\) and \(E^{(2)}\) and compare your result with the corresponding values in Problem 2 of Session 11 obtained using a different partitioning of the Hamiltonian.

(d) Determine \(\Psi^{(n-1)}\) and \(E^{(n)}\) up to order \(n = 6\) in perturbation theory. Compare your result for the correlation energy to the variational CID result obtained in Problem 1 of Session 10.

2. The standard expression for the MP2 correlation energy of a closed-shell system written in terms of two-electron integrals and orbital energies reads

\[ E_\mathrm{MP2} = - \sum_{i,j,s,t} \frac{ (is|jt) \big[ 2 (si|tj) - (sj|ti) \big] }{ \varepsilon_s + \varepsilon_t - \varepsilon_i - \varepsilon_j } \]

where summations run over occupied (indices \(i\) and \(j\)) and unoccupied (indices \(s\) and \(t\)) spatial MOs. Use this formula to determine the MP2 energy and compare to your result from Problem 1 od Session 12.

Session 13#

1. Consider the setup made in Problem 1 of Session 10. With the approach from Problem 1 of Session 2, determine an expression for the one-electron density, \(n(\mathbf{r})\), of the CID wave function in terms of the spatial MOs.

2. Let us define the two-electron density operator as

\[ \hat{n}(\mathbf{r}, \mathbf{r}') = \sum_{j>i}^N \left[ \delta(\mathbf{r} - \mathbf{r}_i) \delta(\mathbf{r}' - \mathbf{r}_j) + \delta(\mathbf{r} - \mathbf{r}_j) \delta(\mathbf{r}' - \mathbf{r}_i) \right] \]

Show that

\[\begin{equation*} \label{eq:den2-oper} n(\mathbf{r}, \mathbf{r}') = \langle \Psi | \hat{n}(\mathbf{r}, \mathbf{r}') | \Psi \rangle \end{equation*}\]

is in agreement with the definition of the one-electron density based on a probabilistic interpretation of the wave function.

3. Consider the setup made in Problem 1 of Session 10. With the approach from Problem 2 of Session 13, determine an expression for the two-electron density, \(n(\mathbf{r}, \mathbf{r}')\), of the CID wave function in terms of the spatial MOs.

Session 14#

Choose from the extra problems.

Contents