[{"problem_text": "Use the $D_0$ value of $\\mathrm{H}_2(4.478 \\mathrm{eV})$ and the $D_0$ value of $\\mathrm{H}_2^{+}(2.651 \\mathrm{eV})$ to calculate the first ionization energy of $\\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\\mathrm{H}_2$ ).", "answer_latex": " 15.425", "answer_number": "15.425", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 13.3", "comment": " ", "solution": ""}, {"problem_text": "Calculate the energy of one mole of UV photons of wavelength $300 \\mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \\mathrm{~kJ} / \\mathrm{mol}$.", "answer_latex": " 399", "answer_number": "399", "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 1.3", "comment": " ", "solution": ""}, {"problem_text": "Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer. ", "answer_latex": " 9.13", "answer_number": "9.13", "unit": " $10^{-35} \\mathrm{~J} \\mathrm{~s}$", "source": "quan", "problemid": " 10.1", "comment": " ", "solution": ""}, {"problem_text": "The ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$ ground electronic state has $D_0=2.4287 \\mathrm{eV}, \\nu_e / c=1405.65 \\mathrm{~cm}^{-1}$, and $\\nu_e x_e / c=23.20 \\mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\\omega_e$ and $\\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$.", "answer_latex": " 2.5151", "answer_number": "2.5151", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 13.5", "comment": " ", "solution": ""}, {"problem_text": "The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an \"atom\" that consists of a positron and an electron.", "answer_latex": " -6.8", "answer_number": " -6.8", "unit": "$\\mathrm{eV}$", "source": "quan", "problemid": " 6.22", "comment": " ", "solution": ""}, {"problem_text": "What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?", "answer_latex": " 14", "answer_number": "14", "unit": " ", "source": "quan", "problemid": " 6.29", "comment": " ", "solution": ""}, {"problem_text": "How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?", "answer_latex": " 15", "answer_number": "15", "unit": " ", "source": "quan", "problemid": " 11.22", "comment": " ", "solution": ""}, {"problem_text": "Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 \u00c5$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 \u00c5$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \\mathrm{~N} / \\mathrm{m}$.", "answer_latex": " 27", "answer_number": "27", "unit": " $\\mathrm{kcal} / \\mathrm{mol}$", "source": "quan", "problemid": " 17.9", "comment": " Angstrom ", "solution": ""}, {"problem_text": "When a particle of mass $9.1 \\times 10^{-28} \\mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \\times 10^{14} \\mathrm{~s}^{-1}$. Find the length of the box.", "answer_latex": " 1.8", "answer_number": "1.8", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 2.13", "comment": " ", "solution": ""}, {"problem_text": "Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.", "answer_latex": " 0.16", "answer_number": "0.16", "unit": " ", "source": "quan", "problemid": " 4.42", "comment": " ", "solution": ""}, {"problem_text": "Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)", "answer_latex": " 0.332", "answer_number": "0.332", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 1.6", "comment": " ", "solution": ""}, {"problem_text": "Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\\alpha$.", "answer_latex": " 54.7", "answer_number": "54.7", "unit": " $^{\\circ}$", "source": "quan", "problemid": " 10.2", "comment": " ", "solution": ""}, {"problem_text": "The AM1 valence electronic energies of the atoms $\\mathrm{H}$ and $\\mathrm{O}$ are $-11.396 \\mathrm{eV}$ and $-316.100 \\mathrm{eV}$, respectively. For $\\mathrm{H}_2 \\mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \\mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \\mathrm{eV}$. For $\\mathrm{H}(g)$ and $\\mathrm{O}(g), \\Delta H_{f, 298}^{\\circ}$ values are 52.102 and $59.559 \\mathrm{kcal} / \\mathrm{mol}$, respectively. Find the AM1 prediction of $\\Delta H_{f, 298}^{\\circ}$ of $\\mathrm{H}_2 \\mathrm{O}(g)$.", "answer_latex": " -59.24", "answer_number": "-59.24", "unit": " $\\mathrm{kcal} / \\mathrm{mol}$", "source": "quan", "problemid": " 17.29", "comment": " ", "solution": ""}, {"problem_text": "Given that $D_e=4.75 \\mathrm{eV}$ and $R_e=0.741 \u00c5$ for the ground electronic state of $\\mathrm{H}_2$, find $U\\left(R_e\\right)$ for this state.", "answer_latex": " -31.95", "answer_number": " -31.95", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.35", "comment": " Angstrom ", "solution": ""}, {"problem_text": "For $\\mathrm{NaCl}, R_e=2.36 \u00c5$. The ionization energy of $\\mathrm{Na}$ is $5.14 \\mathrm{eV}$, and the electron affinity of $\\mathrm{Cl}$ is $3.61 \\mathrm{eV}$. Use the simple model of $\\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \\times 10^{-30} \\mathrm{C} \\mathrm{m}$.]", "answer_latex": " 4.56", "answer_number": " 4.56", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.5", "comment": " Angstrom ", "solution": ""}, {"problem_text": "Find the number of CSFs in a full CI calculation of $\\mathrm{CH}_2 \\mathrm{SiHF}$ using a 6-31G** basis set.", "answer_latex": " 1.86", "answer_number": "1.86", "unit": "$10^{28} $", "source": "quan", "problemid": " 16.1", "comment": " ", "solution": ""}, {"problem_text": "Calculate the ratio of the electrical and gravitational forces between a proton and an electron.", "answer_latex": " 2", "answer_number": "2", "unit": " $10^{39}$", "source": "quan", "problemid": " 6.15", "comment": " ", "solution": ""}, {"problem_text": "A one-particle, one-dimensional system has the state function\r\n$$\r\n\\Psi=(\\sin a t)\\left(2 / \\pi c^2\\right)^{1 / 4} e^{-x^2 / c^2}+(\\cos a t)\\left(32 / \\pi c^6\\right)^{1 / 4} x e^{-x^2 / c^2}\r\n$$\r\nwhere $a$ is a constant and $c=2.000 \u00c5$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 \u00c5$ and $2.001 \u00c5$.", "answer_latex": " 0.000216", "answer_number": "0.000216", "unit": " ", "source": "quan", "problemid": " 1.13", "comment": " Angstrom", "solution": ""}, {"problem_text": "The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \\mathrm{GHz}$, where $1 \\mathrm{GHz} \\equiv 10^9 \\mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.", "answer_latex": " 252.8", "answer_number": " 252.8", "unit": " $\\mathrm{GHz}$", "source": "quan", "problemid": " 6.10", "comment": " Approximated answer", "solution": ""}, {"problem_text": "Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \\mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \\pi \\varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\\psi$ is essentially equal to 1 within the nucleus.\r\n", "answer_latex": " 1.2", "answer_number": "1.2", "unit": " $10^{-8} \\mathrm{eV}$", "source": "quan", "problemid": " 9.9", "comment": " ", "solution": ""}, {"problem_text": "An electron in a three-dimensional rectangular box with dimensions of $5.00 \u00c5, 3.00 \u00c5$, and $6.00 \u00c5$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.", "answer_latex": "7.58", "answer_number": "7.58", "unit": " $10^{14} \\mathrm{~s}^{-1}$", "source": "quan", "problemid": " 3.35", "comment": " Angstrom ", "solution": ""}, {"problem_text": "Do $\\mathrm{HF} / 6-31 \\mathrm{G}^*$ geometry optimizations on one conformers of $\\mathrm{HCOOH}$ with $\\mathrm{OCOH}$ dihedral angle of $0^{\\circ}$. Calculate the dipole moment.", "answer_latex": " 1.41", "answer_number": "1.41", "unit": " $\\mathrm{D}$", "source": "quan", "problemid": " 15.57", "comment": " ", "solution": ""}, {"problem_text": "Frozen-core $\\mathrm{SCF} / \\mathrm{DZP}$ and CI-SD/DZP calculations on $\\mathrm{H}_2 \\mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\\Phi_0$ in the normalized CI-SD wave function.", "answer_latex": " 0.9731", "answer_number": "0.9731", "unit": " ", "source": "quan", "problemid": " 16.3", "comment": " ", "solution": ""}, {"problem_text": "Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\\langle w\\rangle$.", "answer_latex": " 1", "answer_number": "1", "unit": " ", "source": "quan", "problemid": " 5.8", "comment": " ", "solution": ""}, {"problem_text": "Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 \u00c5$.", "answer_latex": " 0.405", "answer_number": "0.405", "unit": " $\\mathrm{~N}$", "source": "quan", "problemid": " 1.31", "comment": " ", "solution": ""}, {"problem_text": "When an electron in a certain excited energy level in a one-dimensional box of length $2.00  \u00c5$ makes a transition to the ground state, a photon of wavelength $8.79 \\mathrm{~nm}$ is emitted. Find the quantum number of the initial state.", "answer_latex": "4", "answer_number": "4", "unit": "", "source": "quan", "problemid": " 2.13", "comment": " Angstrom ", "solution": ""}, {"problem_text": "For a macroscopic object of mass $1.0 \\mathrm{~g}$ moving with speed $1.0 \\mathrm{~cm} / \\mathrm{s}$ in a one-dimensional box of length $1.0 \\mathrm{~cm}$, find the quantum number $n$.", "answer_latex": " 3", "answer_number": "3", "unit": "$10^{26}$", "source": "quan", "problemid": " 2.11", "comment": " ", "solution": ""}, {"problem_text": "For the $\\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \\mathrm{eV}$. Find $\\Delta H_0^{\\circ}$ for $\\mathrm{H}_2(g) \\rightarrow 2 \\mathrm{H}(g)$ in $\\mathrm{kJ} / \\mathrm{mol}$", "answer_latex": " 432.07", "answer_number": "432.07", "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 13.2", "comment": " ", "solution": ""}, {"problem_text": "The contribution of molecular vibrations to the molar internal energy $U_{\\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\\mathrm{m}, \\mathrm{vib}}=R \\sum_{s=1}^{3 N-6} \\theta_s /\\left(e^{\\theta_s / T}-1\\right)$, where $\\theta_s \\equiv h \\nu_s / k$ and $\\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\\mathrm{m}, \\text { vib }}$ at $25^{\\circ} \\mathrm{C}$ of a normal mode with wavenumber $\\widetilde{v} \\equiv v_s / c$ of $900 \\mathrm{~cm}^{-1}$.", "answer_latex": " 0.14", "answer_number": "0.14", "unit": " $\\mathrm{kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 15.39", "comment": " ", "solution": ""}, {"problem_text": "Calculate the magnitude of the spin magnetic moment of an electron.", "answer_latex": " 1.61", "answer_number": "1.61", "unit": " $10^{-23} \\mathrm{~J} / \\mathrm{T}$", "source": "quan", "problemid": " 10.17", "comment": " ", "solution": ""}, {"problem_text": "A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \\mathrm{eV}$, calculate $\\langle V\\rangle$ for the ground state.", "answer_latex": " $3\frac{1}{3}$", "answer_number": "3.333333333", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.29", "comment": " screenshot answer is weird", "solution": ""}, {"problem_text": "For an electron in a certain rectangular well with a depth of $20.0 \\mathrm{eV}$, the lowest energy level lies $3.00 \\mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\\tan \\theta=\\sin \\theta / \\cos \\theta$", "answer_latex": " 0.264", "answer_number": "0.264", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 2.27", "comment": " hint", "solution": ""}, {"problem_text": "Calculate the uncertainty $\\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.", "answer_latex": " 0", "answer_number": "0", "unit": " ", "source": "quan", "problemid": " 7.56", "comment": " ", "solution": ""}]