AZIMUTHAL (ANGULAR MOMENTUM)QUANTUM NUMBER I 11 where the operator is(dd).the eigenfunction is f).and the eigenvalue is(n). Generally.it is implied that wave functions.hence orbitals.are eigenfunctions. 1.23 EIGENVALUES The values of A calculated from the wave equation,Eq.1.17.If the eigenfunction is an orbital,then the eigenvalue is related to the orbital energy. 1.24 THE SCHRODINGER EQUATION FOR THE HYDROGEN ATOM An (eigenvalue)equation,the solutions of which in spherical coordinates are (r,6,p)=R(r)O(6)(p) (1.24) The eigenfunctions o.also called orbitals,are functions of the three variables shown where r is the distance of a point from the origin,and 0 and are the two angles required to locate the point( ee Fig.1.15).For ome e purposes,the spatial or radial and the r part c the Schrodi pendently.Associate par with each eigenfunction (orbit is an eig ue (orb energy).An exact solution of the Schrodinger equation is possible only for the hydrogen atom,or any one-electron system.In many-electron systems wave func- tions are generally approximated as products of modified one-electron functions (orbitals).Each solution of the Schrodinger equation may be distinguished by a set of three quantum numbers,n,1,and m,that arise from the boundary conditions. 1.25 PRINCIPAL QUANTUM NUMBER An integer 1.2.3.....that governs the size of the orbital (wave function)and deter- mic theor and the e la the of the orbita and the farther it extends from the nucleus AZIMUTHAL (ANGULAR MOMENTUM) The quantum number with values of /=0.1.2.....(n-1)that determines the shape of the orbital.The value of I implies particular angular momenta of the electron resulting from the shape of the orbital.Orbitals with the azimuthal quantum numbers 1=0,1.2,and 3 are called s.p.d.and forbitals,respectively.These orbital desig nations are taken from atomic where the words"sharp","principal" "diffuse" and“fnda tal"de cribe l es in atomic spectr ca This ber does not enter into the expression for the energy of an orbital.However,when
where the operator is (d 2 /dx 2 ), the eigenfunction is f(x), and the eigenvalue is (4π2 /λ2 ). Generally, it is implied that wave functions, hence orbitals, are eigenfunctions. 1.23 EIGENVALUES The values of λ calculated from the wave equation, Eq. 1.17. If the eigenfunction is an orbital, then the eigenvalue is related to the orbital energy. 1.24 THE SCHRÖDINGER EQUATION FOR THE HYDROGEN ATOM An (eigenvalue) equation, the solutions of which in spherical coordinates are φ(r, θ, ϕ) R(r) Θ(θ) Φ(ϕ) (1.24) The eigenfunctions φ, also called orbitals, are functions of the three variables shown, where r is the distance of a point from the origin, and θ and ϕ are the two angles required to locate the point (see Fig. 1.15). For some purposes, the spatial or radial part and the angular part of the Schrödinger equation are separated and treated independently. Associated with each eigenfunction (orbital) is an eigenvalue (orbital energy). An exact solution of the Schrödinger equation is possible only for the hydrogen atom, or any one-electron system. In many-electron systems wave functions are generally approximated as products of modified one-electron functions (orbitals). Each solution of the Schrödinger equation may be distinguished by a set of three quantum numbers, n, l, and m, that arise from the boundary conditions. 1.25 PRINCIPAL QUANTUM NUMBER n An integer 1, 2, 3, . . . , that governs the size of the orbital (wave function) and determines the energy of the orbital. The value of n corresponds to the number of the shell in the Bohr atomic theory and the larger the n, the higher the energy of the orbital and the farther it extends from the nucleus. 1.26 AZIMUTHAL (ANGULAR MOMENTUM) QUANTUM NUMBER l The quantum number with values of l 0, 1, 2, . . . , (n 1) that determines the shape of the orbital. The value of l implies particular angular momenta of the electron resulting from the shape of the orbital. Orbitals with the azimuthal quantum numbers l 0, 1, 2, and 3 are called s, p, d, and f orbitals, respectively. These orbital designations are taken from atomic spectroscopy where the words “sharp”, “principal”, “diffuse”, and “fundamental” describe lines in atomic spectra. This quantum number does not enter into the expression for the energy of an orbital. However, when AZIMUTHAL (ANGULAR MOMENTUM) QUANTUM NUMBER l 11 c01.qxd 5/17/2005 5:12 PM Page 11
2 ATOMIC ORBITAL THEORY electrons are placed in orbitals,the energy of the orbitals(and hence the energy of the electrons in them)is affected so that orbitals with the same principal quantum number n may vary in energy. Example.An electron in an orbital with a principal quantum number of n=2 can take onI values of 0 and 1,corresponding to 2s and 2p orbitals,respectively.Although these orbitals have the same principal quantum number and,therefore,the same energy when calculated for the single electron hydrogen atom,for the many-electron whe re e ctron ractions become important,the 2p orbitals ar 1.27 MAGNETIC QUANTUM NUMBER This is the quantum number having values of the azimuthal quantum number from +to-/that determines the orientation in space of the orbital angular momentum; it is represented by m Example.When n=2 and I=1 (the p orbitals).m may thus have values of +1.0. -1,corresponding to three 2p orbitals (see Sect.1.35).When n=3 and I=2,m,has the values of +2,+1,0,-1,-2 that describe the five 3d orbitals (see Sect.1.36). 1.28 DEGENERATE ORBITALS Orbitals having equal energies,for example,the three 2p orbitals. 1.29 ELECTRON SPIN QUANTUM NUMBER m, ntum of the electron due to the fac that the ele on its s spinning:it is usually designated by m,and may only have the value of 1/2 or-1/2 1.30 s ORBITALS Spherically symmetrical orbitals:that is,is a function of R(r)only.For s orbitals, 1=0 and,therefore,electrons in such orbitals have an orbital magnetic quantum number m equal to zero. 1.31 1s ORBITAL The lowest-energy orbital of any atom,characterized by n=1.I=m=0.It corre- sponds to the fundamental wave and is characterized by spherical symmetry and no
electrons are placed in orbitals, the energy of the orbitals (and hence the energy of the electrons in them) is affected so that orbitals with the same principal quantum number n may vary in energy. Example. An electron in an orbital with a principal quantum number of n 2 can take on l values of 0 and 1, corresponding to 2s and 2p orbitals, respectively. Although these orbitals have the same principal quantum number and, therefore, the same energy when calculated for the single electron hydrogen atom, for the many-electron atoms, where electron–electron interactions become important, the 2p orbitals are higher in energy than the 2s orbitals. 1.27 MAGNETIC QUANTUM NUMBER ml This is the quantum number having values of the azimuthal quantum number from l to l that determines the orientation in space of the orbital angular momentum; it is represented by ml . Example. When n 2 and l 1 (the p orbitals), ml may thus have values of 1, 0, 1, corresponding to three 2p orbitals (see Sect. 1.35). When n 3 and l 2, ml has the values of 2, 1, 0, 1, 2 that describe the five 3d orbitals (see Sect. 1.36). 1.28 DEGENERATE ORBITALS Orbitals having equal energies, for example, the three 2p orbitals. 1.29 ELECTRON SPIN QUANTUM NUMBER ms This is a measure of the intrinsic angular momentum of the electron due to the fact that the electron itself is spinning; it is usually designated by ms and may only have the value of 1/2 or 1/2. 1.30 s ORBITALS Spherically symmetrical orbitals; that is, φ is a function of R(r) only. For s orbitals, l 0 and, therefore, electrons in such orbitals have an orbital magnetic quantum number ml equal to zero. 1.31 1s ORBITAL The lowest-energy orbital of any atom, characterized by n 1, l ml 0. It corresponds to the fundamental wave and is characterized by spherical symmetry and no 12 ATOMIC ORBITAL THEORY c01.qxd 5/17/2005 5:12 PM Page 12
2s ORBITAL 13 sented by a p jection of a sphere (a circle)surrounding the nucleus Erample.The numerical probability of finding the hydrogen electron within spheres of various radii from the nucleus is shown in Fig.1.3la.The circles represent con- tours of probability on a plane that bisects the sphere.If the contour circle of 0.95 probability is chosen.the electron is 19 times as likely to be inside the correspon- ding sphere with a radius of 1.7A as it is to be outside that sphere.The circle that is usually drawn, nt the is mply that t unspeci probabi of finding the electron in a sphere.of which the probability 0.95 0.9 07 0.5 0.3 0.4p11 20 radius(A) Figure 1.31.()The probability contours and radii for the hydrogen atom,the probability at the nucleus is zero.(b)Representation of the 1s orbital. 1.32 2s ORBITAL The spherically symmetrical orbital having one spherical nodal surface,that is.a sur face on which the probability of finding an electron is zero.Electrons in this orbital have the principal quantum number n=2,but have no angular momentum,that is, 1=0、m=0. Example.Figure 1.32 shows the probability distribution of the 2s electron as a cross the sphe ical 2s simple circle
nodes. It is represented by a projection of a sphere (a circle) surrounding the nucleus, within which there is a specified probability of finding the electron. Example. The numerical probability of finding the hydrogen electron within spheres of various radii from the nucleus is shown in Fig. 1.31a. The circles represent contours of probability on a plane that bisects the sphere. If the contour circle of 0.95 probability is chosen, the electron is 19 times as likely to be inside the corresponding sphere with a radius of 1.7 Å as it is to be outside that sphere. The circle that is usually drawn, Fig. 1.31b, to represent the 1s orbital is meant to imply that there is a high, but unspecified, probability of finding the electron in a sphere, of which the circle is a cross-sectional cut or projection. 1.32 2s ORBITAL The spherically symmetrical orbital having one spherical nodal surface, that is, a surface on which the probability of finding an electron is zero. Electrons in this orbital have the principal quantum number n 2, but have no angular momentum, that is, l 0, ml=0. Example. Figure 1.32 shows the probability distribution of the 2s electron as a cross section of the spherical 2s orbital. The 2s orbital is usually drawn as a simple circle of arbitrary diameter, and in the absence of a drawing for the 1s orbital for comparison, 2s ORBITAL 13 1.2 1.6 2.0 0.95 0.9 0.8 0.7 0.5 0.4 0.8 0.3 0.1 (a) (b) probability radius (Å) Figure 1.31. (a) The probability contours and radii for the hydrogen atom, the probability at the nucleus is zero. (b) Representation of the 1s orbital. c01.qxd 5/17/2005 5:12 PM Page 13
4 ATOMIC ORBITAL THEORY 95%contour line Figure 1.32.Probability distribution for the 2s orbital the two would be indistinguishable despite the larger size of the 2s orbital and the fact that there is a nodal surface within the 2s sphere that is not shown in the simple circu- lar representation. 1.33 p ORBITALS These are orbitals with an angular cipal quantum num ding to m=+1,0.-1.In a useful convention,these three orbitals,which are mutually perpendicular to each other,are oriented along the three Cartesian coordi- nate axes and are therefore designated as pp.and pThey are characterized by having one nodal plane. 1.34 NODAL PLANE OR SURFACE n is zero.It h noda and is associated with a change in sign of the wave function
the two would be indistinguishable despite the larger size of the 2s orbital and the fact that there is a nodal surface within the 2s sphere that is not shown in the simple circular representation. 1.33 p ORBITALS These are orbitals with an angular momentum l equal to 1; for each value of the principal quantum number n (except for n 1), there will be three p orbitals corresponding to ml 1, 0, 1. In a useful convention, these three orbitals, which are mutually perpendicular to each other, are oriented along the three Cartesian coordinate axes and are therefore designated as px , py , and pz . They are characterized by having one nodal plane. 1.34 NODAL PLANE OR SURFACE A plane or surface associated with an orbital that defines the locus of points for which the probability of finding an electron is zero. It has the same meaning in three dimensions that the nodal point has in the two-dimensional standing wave (see Sect. 1.7) and is associated with a change in sign of the wave function. 14 ATOMIC ORBITAL THEORY nodal contour region 95% contour line Figure 1.32. Probability distribution ψ2 for the 2s orbital. c01.qxd 5/17/2005 5:12 PM Page 14
2p ORBITALS 15 1.35 2p ORBITALS The set of three degenerate(equal energy)atomic orbitals having the principal quan tum number(n)of 2,an azimuthal quantum number(/)of 1,and magnetic quantum numbers (m)of +1,0,or-1.Each of these orbitals has a nodal plane. Erample.The 2p orbitals are usually depicted so as to emphasize their angular dependence,that is,R(r)is assumed constant,and hence are drawn for co e as a plana section through a th ree-dim .The ensionalrepresentaio of planar cross section of the P:orbita then becomesa pair of circles touching at the origin (Fig.1.35a).In this figure the wave function (without proof)is =e()=(V6/2)cos 0.Since cos6,in the region 90<0<270,is negative,the top circle is positive and the bottom circle nega- tive.However.the physically significant property of an orbital o is its square.2; the plot of2=日2（θ）=3/2cos20 for the p,orbital is shown in Fig.1.35b.which the volume of pace in which there ig probab of finding tho rbital.The shape of this orbital is the fan nilia ongated 0eiedwid bbell ith both obes sign. comm drawing ction,I nodal property,(Fig.1.35c).If the function R(r)is included,the oval-shaped con- tour representation that results is shown in Fig.1.35d,where 2(p,)is shown as a cut in the yz-plane (d Figure 1.35.(a)T o the p.orbital:( (B)the s 2p orbitals;and (d)c ng the ra ace of
1.35 2p ORBITALS The set of three degenerate (equal energy) atomic orbitals having the principal quantum number (n) of 2, an azimuthal quantum number (l) of 1, and magnetic quantum numbers (ml ) of 1, 0, or 1. Each of these orbitals has a nodal plane. Example. The 2p orbitals are usually depicted so as to emphasize their angular dependence, that is, R(r) is assumed constant, and hence are drawn for convenience as a planar cross section through a three-dimensional representation of Θ(θ)Φ(ϕ). The planar cross section of the 2pz orbital, ϕ 0, then becomes a pair of circles touching at the origin (Fig. 1.35a). In this figure the wave function (without proof ) is φ Θ(θ) (6/2)cos θ. Since cos θ, in the region 90° θ 270°, is negative, the top circle is positive and the bottom circle negative. However, the physically significant property of an orbital φ is its square, φ2; the plot of φ2 Θ2(θ) 3/2 cos2 θ for the pz orbital is shown in Fig. 1.35b, which represents the volume of space in which there is a high probability of finding the electron associated with the pz orbital. The shape of this orbital is the familiar elongated dumbbell with both lobes having a positive sign. In most common drawings of the p orbitals, the shape of φ2, the physically significant function, is retained, but the plus and minus signs are placed in the lobes to emphasize the nodal property, (Fig. 1.35c). If the function R(r) is included, the oval-shaped contour representation that results is shown in Fig. 1.35d, where φ2(pz) is shown as a cut in the yz-plane. 2p ORBITALS 15 (d) y z 0.50 1.00 1.50 90° − 180° 270° 0° units of Bohr radii + (a) (b) (c) px pz py + + + − − − Figure 1.35. (a) The angular dependence of the pz orbital; (b) the square of (a); (c) the common depiction of the three 2p orbitals; and (d) contour diagram including the radial dependence of φ. c01.qxd 5/17/2005 5:12 PM Page 15