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Lecture 1 THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 1) Theoretical description of the hydrogen atom: Schrödinger equation: H = E H H is the Hamiltonian operator (kinetic + potential energy of an electron) E – the energy of an electron whose distribution in space is described by - wave function * is proportional to the probability of finding the electron in a given point of space If is normalized, the following holds: The part of space where the probability to find electron is non-zero is called orbital When we plot orbitals, we usually show the smallest volume of space where this probability is 90%. Comments on the Schrodinger equation :

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2) Solution of the Schrödinger equation for the hydrogen atom. Separation of variables r, For the case of the hydrogen atom the Schrödinger equation can be solved exactly when variables r, are separated like = R(r) Three integer parameters appear in the solution, n (principal quantum number), while solving R-equation l (orbital quantum number), while solving - equation m l (magnetic quantum number),while solving -equation n = 1, 2, … ∞; defines energy of an electron l = 0, … n-2, n-1 (total n values); defines shape of an orbital: 0-s, 1-p, 2-d, 3-f,... m l = -l, -l+1, …, 0, …, l-1, l (total 2l+1 values); defines spatial orientation The number of all possible combinations of l and m l for a given n is Solution of the Schrodinger equation for the case of the hydrogen atom:

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Plot of the hydrogen atom orbitals s orbitals p orbitals d orbitals

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3) The hydrogen atom orbitals Radial components of the hydrogen atom wavefunctions look as follows: R(r) = for n = 1, l = 0 1s orbital R(r) = for n = 2, l = 02s orbital R(r) = for n = 2, l = 1, m l = 02p z orbital R(r) = n = 3, l = 03s orbital R(r) = n = 3, l = 1, m l = 03p z orbital Here a 0 =, the first orbit radius (0.529Å). Z is the nuclear charge (+1)

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4) Radial components of the hydrogen atom orbitals The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l-1 in total 1s 2s 2p node

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5) Radial probability function Radial probability function is defined as [r R(r)] 2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: 1s 2s 2p

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6) Angular components Angular components of the hydrogen atom wave functions, for l = 0, m l = 0; = s orbitals for l = 1, m l = 0; = p z orbitals for l = 2, m l = 0; = d z2 orbitals For each orbital we have l angular nodal surfaces Therefore, together with radial nodes we have in total (n-l-1)+l = n-1 nodes for each orbital Hydrogen-like orbitals (animated):

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7) Plot of the hydrogen atom orbitals Orbitals are called gerade (g) if they have center of symmetry or ungerade (u) if they do not have it Examples of orbitals which are gerade: s, d. Orbitals which are ungerade: p.

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8) Allowed energies of an electron in the hydrogen atom E = Note: E does not depend on either l or m l In other words, for each given n s, p, d etc orbitals of the hydrogen atom are of the same energy (degenerate) Spectrum of hydrogen atom: 1s 2s, 2p x, 2p y, 2p z 3s, 3p, 3d 4s, 4p, 4d, 4f

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Summary Exact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation; Three quantum numbers n, l, m l appear as integer parameters while solving the equation; Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (m l ), number of nodal surfaces (n-1) and symmetry (g, u).

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