how to use perturbation theory


Along directions for which \(\cos{\theta}\) is negative (to the right in Figure 4.2 a), this potential becomes large and positive as the distance \(r\) of the electron from the nucleus increases; for bound states such as the \(2s\) and \(2p\) states discussed earlier, such regions are classically forbidden and the wave function exponentially decays in this direction. Try to make sure you can do the algebra, but also make sure you understand how we are using the first-order perturbation equations. Perturbation is used to find the roots of an algebraic equation that differs slightly from one for which the roots are known. \left(\begin{array}{cc}C \\ D\end{array}\right)\\ As long as the zeroth-order energy is not degenerate with \(E^{(0)}\) (or, that the zeroth-order states have been chosen as discussed earlier to cause there to no contribution to \(\psi^{(n)}\) from such degenerate states), the above equation can be solved for the expansion coefficients \(\langle \psi_J^{(0)}|\psi^{(n)}\rangle\), which then define \(\psi^{(n)}\). Perturbation theory is widely used when the problem at hand does not have a known exact solution, but can be expressed as a "small" change to a known solvable problem. In particular, second- and third-order approximations are easy to compute and notably improve accuracy. Because \(H^{(0)}\) is one-electron additive, its eigenfunctions consist of products of eigenfunctions of the operator, \[h^{(0)}=\frac{1}{2}\nabla^2-\frac{3}{r}+V_{\rm HF}(r)\], \(V_{\rm HF}(\textbf{r}_i)\) offers an approximation to the true \(1/r_{i,j}\) Coulomb interactions expressed in terms of a “smeared-out” electron distribution interacting with the electron at ri. Perturbation theory is a collection of methods for the systematic analysis of the global behavior of solutions to differential and difference equations. A fundamental assumption of perturbation theory is that the wave functions and energies for the full Hamiltonian \(H\) can be expanded in a Taylor series involving various powers of the perturbation parameter \(\lambda\). We now use matrix perturbation theory to compute the covariance of based on this zero approximation. Other examples occur in differential equations. 2. In the wave function calculation, we will only compute the contribution to \(\psi\) made by \(\psi^{(0)}_2\) (this is just an approximation to keep things simple in this example). complex, we will use the same notation, jϕn n2W deg, for this new choice of the degenerate, unperturbed eigenvectors. 2. A very good treatment of perturbation theory is in Sakurai’s book –J.J. remains valid, but the summation index \(J\) is now restricted to exclude any members of the zeroth-order states that are degenerate with \(\psi^{(0)}\). To obtain the expression for the second-order correction to the energy of the state of interest, one returns to, \[H^{(0)} \psi^{(2)} + V \psi^{(1)} = E^{(0)} \psi^{(2)} + E^{(1)} \psi^{(1)} + E^{(2)} \psi^{(0)}\], Multiplying on the left by the complex conjugate of \(\psi^{(0)}\) and integrating yields, \[\langle\psi^{(0)}|H^{(0)}|\psi^{(2)}\rangle + \langle\psi^{(0)}|V|\psi^{(1)}\rangle = E^{(0)} \langle\psi^{(0)}|\psi^{(2)}\rangle + E^{(1)} \langle\psi^{(0)}|\psi^{(1)}\rangle + E^{(2)} \langle\psi^{(0)}|\psi^{(0)}\rangle.\], The intermediate normalization condition causes the fourth term to vanish, and the first and third terms cancel one another. 0. perturbation theory to account for these effects. In QM, it is said that perturbation theory can be used in the case in which the total Hamiltonian is a sum of two parts, one whose exact solution is known and an extra term that contains a small parameter, λ say. \left(\begin{array}{cc} \(E^{(3)}\) (and higher \(E^{(n)}\)) are then determined from \(\langle\psi_0|V|\psi_{n-1}\rangle = E^{(n)}\) and the expansion coefficients of \(\psi^{(2)}\) {\(\langle |\psi^{(2)}\rangle\)} are determined from the above equation with \(n = 2\). $\begingroup$ Any specific kind of perturbation theory? There will, however, be second-order Stark splittings, in which case we need to examine the terms that arise in the formula, \[E^{(2)}=\sum_J\frac{|\langle\psi^{(0)}|V|psi^{(0)}\rangle|^2}{E^{(0)}-E^{(0)}_J}\], For a zeroth-order state \(Y_{J,M}\), only certain other zeroth-order states will have non-vanishing coupling matrix elements . on the zeroth-order states. It allows us to get good approximations for system where the Eigen values cannot be easily determined. An electron moves in the potential well P ( x) = − δ for − a < x < 0 and P ( x) = δ for 0 < x < a (Fig. In the expression for \(E^{(1)} = \langle\psi^{(0)}|V|\psi^{(0)}\rangle\), the product \(\psi^{(0)}{}^*\psi^{(0)}\) is an even function under reflection of \(x\) through the midpoint \(x = \dfrac{L}{2}\); in fact, this is true for all of the particle-in-a-box wave functions. Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry, Integrated by Tomoyuki Hayashi (UC Davis). As noted earlier, this means that one must solve \(H^{(0)} \psi^{(0)}_J = E^{(0)}J \psi^{(0)}_J\) not just for the zeroth-order state one is interested in (denoted \(\psi^{(0)}\) above), but for all of the other zeroth-order states \(\{\psi^{(0)}_J\}\). 2. Again, allow me to do the algebra and see if you can follow. If one were to try to solve \(\langle \psi_J^{(0)}|\psi^{(1)}\rangle + \langle\psi_J^{(0)} |V|\psi_0\rangle = E^{(0)} \langle \psi_J^{(0)}|\psi^{(1)}\rangle\) without taking this extra step, the \(\langle\psi_J^{(0)} |\psi^{(1)}\rangle\) values for those states with \(= E^{(0)}\) could not be determined because the first and third terms would cancel and the equation would read \(\langle \psi_J^{(0)}|V|\psi^{(0)}\rangle = 0\). In most practical applications of quantum mechanics to molecular problems, one is faced with the harsh reality that the Schrödinger equation pertinent to the problem at hand cannot be solved exactly. A –rst-order perturbation theory and linearization deliver the same output. \], where \(z\) is taken to be the direction of the electric field. The diagonal elements of the electric-dipole operator, \[\langle ​Y_{J,M}(\theta,\phi)\chi_\nu(R)\psi_e(r|R)|V|Y_{J,M}(\theta,\phi)\chi_\nu(R)\psi_e(r|R) \rangle\], vanish because the vibrationally averaged dipole moment, which arises as, \[\langle\mu\rangle = \langle ​\chi_\nu(R)\psi_e(r|R)| e\sum_n Z_n \textbf{R}_n-e\sum_i \textbf{r}_i |\chi_\nu(R)\psi_e(r|R) \rangle\], is a vector quantity whose component along the electric field is \(\langle \mu\rangle \cos(\theta)\) (again taking the field to lie along the \(z\)-direction). For example, expanding \(\psi^{(1)}\) in this manner gives: Now, the unknowns in the first-order equation become \(E^{(1)}\) and the expansion coefficients. None of these problems, even the classical Newton’s equation for the sun, earth, and moon, have ever been solved exactly. This result also suggests that the polarizability of conjugated polyenes should vary non-linearly with the length of the conjugated chain. Another example of Stark effects in degenerate cases arises when considering how polar diatomic molecules’ rotational energies are altered by an electric field. In such cases, it is usual to write the decomposition of \(H\) as. This process can then be continued to higher and higher order. – 1 eV in Figure 4.2 a) and then decreases monotonically as r increases. If one is dealing with a degenerate state of a centro-symmetric system, things are different. The nth order energy is obtained by multiplying this equation on the left by \(\psi^{(0)}{}^*\) and integrating over the relevant coordinates (and using the fact that \(\psi^{(0)}\) is normalized and the intermediate normalization condition \(\langle\psi^{(0)}|\psi_m\rangle = 0\) for all \(m > 0\)): \[\langle\psi^{(0)}|V|\psi^{(n-1)}\rangle = E^{(n)}.\], This allows one to recursively solve for higher and higher energy corrections once the various lower-order wave functions \(\psi^{(n-1)}\) are obtained. \left(\begin{array}{cc} its energy levels and eigenstates For such a polarized superposition wave function, there should be a net dipole moment induced in the system. In fact, this potential approaches \(-\infty\) as \(r\) approaches \(\infty\) as we see in the left portion of Figure 4.2 a. finally, the \(J = J’\) terms will vanish because of the inversion symmetry (\(\cos\theta\) is odd under inversion but \(|Y_{J,M}|^2\) is even). In Figure 4.1 we show the \(n = 1\) and \(n = 2\) zeroth-order functions as well as the superposition function formed when the zeroth-order \(n = 1\) and first-order \(n = 1\) functions combine. We can evaluate this dipole moment by computing the expectation value of the dipole moment operator: \[\mu_{induced}= - e \int\psi^*\left(x-\frac{L}{2}\right)\psi dx\]. As we discussed earlier, an electron moving in a quasi-linear conjugated bond framework can be modeled as a particle in a box. The two corrected zeroth-order wave functions corresponding to these two shifted energies are, \[\psi^{(0)}_{\pm}=\frac{1}{\sqrt{2}}[2s\mp 2p_z]\]. So, the wave function through first order (i.e., the sum of the zeorth- and first-order pieces) is, \[\psi^{(0)}+\psi^{(1)}=\sqrt{\frac{2}{L}}\sin\left(\frac{\pi x}{L}\right) + \frac{32mL^3e\varepsilon}{27\hbar^2\pi^4}\sqrt{\frac{2}{L}}\sin\left(\frac{2\pi x}{L}\right)\]. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The zeroth-order equation simply instructs us to solve the model Schrödinger equation to obtain the zeroth-order wave function \(\psi^{(0)}\) and its zeroth-order energy \(E^{(0)}\). The limitation that \(M\) must equal \(M’\) arises, as above, because the perturbation contains no terms dependent on the variable \(\phi\). r = s ( θ, t) ∼ s 0 ( t) + ϵ s 1 ( θ, t) + ϵ 2 s 2 ( θ, t) + O ( ϵ 3). In the first-order equation, the unknowns are \(\psi^{(1)}\) and \(E^{(1)}\) (recall that \(V\) is assumed to be known because it is the difference between the Hamiltonian one wants to solve and the model Hamiltonian \(H^{(0)}\)). Perturbation theory comprises mathematical strategies for finding an approximate strategy to a trouble, by way of starting from the precise solution of a related, less difficult problem. H‰l•{PÒéÇ~*x…_hj¨Ü4–ÐCÊ8Tæ "¶©¤ÆLÉiFÝl³µ3 (?Ì4­T@OJéJ®[ž.ë5/ÛEÜìvΞÂ[æì9¥¹s´³gæPS{ÎÌö̼ïoÞ÷ç7Ïçûƒ!`0—_a®Ž]aÿæâ¸`Èâä_¼žûd(ýgàÝæÿí|úÁŠ×y. The \(J_{1s,1s}+2J_{1s,2s}\) terms describe the Coulombic repulsions among the three electrons. It vanishes in the two specific cases mentioned because \(\psi\) is either even or odd under the inversion symmetry, but the product \(\psi^* \psi\) is even, and the dipole operator is odd, so the integrand is odd and thus the integral vanishes. The RSPT equations can be solved recursively to obtain even high-order energy and wave function corrections: Although modern quantum mechanics uses high-order perturbation theory in some cases, much of what the student needs to know is contained in the first- and second- order results to which I will therefore restrict our further attention. The following text is an example of how to use the ideas set out in the perturbation theory writeup. So RSPT assumes that one has already chosen the degenerate sets of zeroth-order states to make \(\langle \psi_J^{(0)}|V| \psi_K^0\rangle = 0\) for \(K \ne J\). Since \(H^{(0)}\) is a Hermitian operator, it has a complete set of such eigenfunctions, which we label \(\{\psi^{(0)}k\}\) and {E^{(0)}_k}. ,of \(\cos(\theta)\)). Using perturbation theory in such a simple or in a somewhat more complicated form enables us to find the answers to a whole set of problems in radiation theory (Heitler, 1947; Berestetskii, Lifshitz and Pitaevskii, 1971). For example, the \(2s\) and \(2p\) states of the hydrogen atom are degenerate, so, to apply perturbation theory one has to choose specific combinations that diagonalize the perturbation. Use first-order perturbation theory to compute the first four energy levels. The Schrödinger equation for the two electrons moving about the He nucleus: The Schrödinger equation for the two electrons moving in an \(H_2\) molecule even if the locations of the two nuclei (labeled A and B) are held clamped as in the Born-Oppenheimer approximation: \(\psi^{(0)}\) and \(E^{(0)}\) and \(V\) are used to determine \(E^{(1)}\) and \(\psi^{(1)}\) as outlined above. But, first, let’s consider an example problem that illustrates how perturbation theory is used in a more quantitative manner. 320 0 obj << /Linearized 1 /O 322 /H [ 1788 3825 ] /L 800913 /E 166218 /N 83 /T 794394 >> endobj xref 320 69 0000000016 00000 n 0000001731 00000 n 0000005613 00000 n 0000005831 00000 n 0000006141 00000 n 0000006248 00000 n 0000006429 00000 n 0000006537 00000 n 0000006589 00000 n 0000006611 00000 n 0000007328 00000 n 0000008068 00000 n 0000008432 00000 n 0000008587 00000 n 0000008810 00000 n 0000029538 00000 n 0000029784 00000 n 0000029986 00000 n 0000030189 00000 n 0000030404 00000 n 0000047764 00000 n 0000048182 00000 n 0000049146 00000 n 0000049846 00000 n 0000050059 00000 n 0000050372 00000 n 0000065848 00000 n 0000066192 00000 n 0000066671 00000 n 0000066895 00000 n 0000067276 00000 n 0000088258 00000 n 0000088471 00000 n 0000088674 00000 n 0000106073 00000 n 0000106352 00000 n 0000106897 00000 n 0000107389 00000 n 0000122733 00000 n 0000123067 00000 n 0000123325 00000 n 0000123347 00000 n 0000123951 00000 n 0000123973 00000 n 0000124523 00000 n 0000136299 00000 n 0000136493 00000 n 0000137184 00000 n 0000137537 00000 n 0000137963 00000 n 0000138155 00000 n 0000159050 00000 n 0000159118 00000 n 0000159494 00000 n 0000159680 00000 n 0000159702 00000 n 0000160266 00000 n 0000160288 00000 n 0000160830 00000 n 0000160852 00000 n 0000161346 00000 n 0000161368 00000 n 0000161940 00000 n 0000161962 00000 n 0000162436 00000 n 0000162515 00000 n 0000163571 00000 n 0000001788 00000 n 0000005590 00000 n trailer << /Size 389 /Info 309 0 R /Root 321 0 R /Prev 794383 /ID[<80e4976f4d4a68bc24a5413a63c0b3eb><80e4976f4d4a68bc24a5413a63c0b3eb>] >> startxref 0 %%EOF 321 0 obj << /Type /Catalog /Pages 311 0 R >> endobj 387 0 obj << /S 5229 /Filter /FlateDecode /Length 388 0 R >> stream In computing this integral, we neglect the term proportional to \(E^{(2)}\) because we are interested in only the term linear in \(\varepsilon\) because this is what gives the dipole moment. 1. This choice ensures that the matrix E(1) n′n is diagonal, the goal of first-order perturbation theory. Stationary perturbation theory 65 Now, the operator W may be written in matrix form in the | E0,ai basis as W11 W12 W21 W22 so that equations (29) and (31) may be written as the matrix equation W µ α1 α2 = E1 µ α1 α2 The characteristic equation det(W − E1I) = 0 may then be solved in order to find the two eigenvalues and eigenstates. \[\mu_{induced}= - e \int_0^L(\psi^{(0)}+\psi^{(1)})^*\left(x-\frac{L}{2}\right)(\psi^{(0)}+\psi^{(1)}) dx\], \[= - e \int_0^L​ \psi^{(0)}{}^*\left(x-\frac{L}{2}\right)\psi^{(0)} dx - e \int_0^L​ \psi^{(1)}{}^*\left(x-\frac{L}{2}\right)\psi^{(0)} dx\], \[= - e \int_0^L \psi^{(0)}{}^*\left(x-\frac{L}{2}\right)\psi^{(1)} dx - e \int_0^L \psi^{(1)}{}^*\left(x-\frac{L}{2}\right)\psi^{(1)} dx\]. It should be noted that there are problems which cannot be solved using perturbation theory, First, one decomposes the true Hamiltonian H into a so-called zeroth-order part H ( 0) (this is the Hamiltonian of the model problem used to represent the real system) and the difference ( H − H ( 0) ), which is called the perturbation and usually denoted V: H = H ( 0) + V. This has been achieved by a more physical choice of the 4 Watch the recordings here on Youtube! One of these states will be the one we are interested in studying (e.g., we might be interested in the effect of an external electric field on the \(2s\) state of the hydrogen atom), but, as will become clear soon, we actually have to find the full set of {\(\psi^{(0)}_k\)} and {\(E^{(0)}_k\)} (e.g., we need to also find the \(1s, 2p, 3s, 3p, 3d,\) etc. However, in directions along which \(\cos{\theta}\) is positive, the potential is negative and strongly attractive for small-r (i.e., near the nucleus), then passes through a maximum (e.g., near \(x = -2\) in Figure 4.2 a) at, \[r_{\rm max}=\sqrt{\frac{Z}{e\textbf{E}\cos\theta}}\], \[V(r_{\rm max})=-2\sqrt{e\textbf{E}\cos\theta}\]. This potential, which is written in terms of Coulomb integrals similar to those we discussed earlier as well as so-called exchange integrals that we will discuss in Chapter 6, is designed to approximate the interaction of an electron at location \(\textbf{r}_i\) with the other electrons in the atom or molecule. The first integral can be evaluated using the following identity with \(a = \dfrac{\pi}{L}\): \[\int_0^L\sin^2(ax)dx=\frac{x^2}{4}-\frac{x\sin(2ax)}{4a}-\frac{x\cos(2ax)}{8a^2}\Bigg|^L_0=\frac{L^2}{4}\], The second integral can be evaluated using the following identity with \(\theta =\frac{\pi x}{L}\). Similarly, the energy is written as a sum of terms of increasing order. 0. The zeroth, first, and second-order such equations are given below: \[H^{(0)} \psi^{(0)} = E^{(0)} \psi^{(0)},\], \[H^{(0)} \psi^{(1)} + V \psi^{(0)} = E^{(0)} \psi^{(1)} + E^{(1)} \psi^{(0)}\], \[H^{(0)} \psi^{(2)} + V \psi^{(1)} = E^{(0)} \psi^{(2)} + E^{(1)} \psi^{(1)} + E^{(2)} \psi^{(0)}.\], It is straightforward to see that the nth order expression in this sequence of equations can be written as, \[H^{(0)} \psi^{(n)} + V \psi^{(n-1)} = E^{(0)} \psi^{(n)} + E^{(1)} \psi^{(n-1)} + E^{(2)} \psi^{(n-2)} + E^{(3)} \psi^{(n-3)} + \cdots + E^{(n)} \psi^{(0)}.\]. ϕ. It's a much better approximation using tree-level scattering results at high energies using the running coupling than to use the (quasi-)onshell scheme from low-energy QED (the running coupling at a scale around the Z-mass is 1/128 rather than 1/137). Thinking of \(\cos(\theta)\) as \(x\), so \(\sin(\theta) d\theta\) is \(dx\), the integrals, \[\langle ​Y_{J,M}(\theta,\phi)|\cos\theta|Y_{J,M}(\theta,\phi)\rangle=\int Y_{J,M}^*(\theta,\phi) \cos\theta ​Y_{J,M}(\theta,\phi) \sin\theta d\theta d\phi=\int Y_{J,M}^*(\theta,\phi) x ​Y_{J,M}(\theta,\phi) dxd\phi=0\], because \(|Y_{J,M}|^2\) is an even function of \(x\) (i.e. k + ..., E. k = E. k + Ç«E. ​\end{array}\right) We substitute this formal series into the perturbed equation and appeal to (5.1) by successively To solve the first-order and higher-order equations, one expands each of the corrections to the wave function \(\psi\) of interest in terms of the complete set of wave functions of the zeroth-order problem \(\{\psi^{(0)}_J\}\). The zeroth-order wave functions appropriate to such cases are given by, \[\psi=Y_{J,M}(\theta,\phi)\chi_\nu(R)\psi_e(r|R)\], where the spherical harmonic \(Y_{J,M}(\theta,\phi)\) is the rotational wave function, \(\chi_\nu(R)\) is the vibrational function for level \(\nu\), and \(\psi_e(r|R)\) is the electronic wave function. - \left(\frac{L^2}{9\pi^2}\cos\left(\dfrac{3\pi x}{L}\right)+ \frac{Lx}{3\pi}\sin\left(\dfrac{3\pi x}{L}\right)\right) \Bigg|^L\right] \], \[=\frac{-2L^2}{2\pi^2} -\frac{-2L^2}{18\pi^2} = \frac{L^2}{9\pi^2} -\frac{L^2}{\pi^2} = - \frac{-8L^2}{9\pi^2}.\], Making all of these appropriate substitutions we obtain: Let us translate the above statement into a precise mathematical framework. The bottom line is that the total potential with the electric field present violates the assumptions on which perturbation theory is based. It says the first-order correction to the energy \(E^{(0)}\) of the unperturbed state can be evaluated by computing the average value of the perturbation with respect to the unperturbed wave function \(\psi^{(0)}\). The quantity inside the integral is the electric dipole operator, so this integral is the dipole moment of the molecule in the absence of the field. The perturbation theory approach provides a set of analytical expressions for generating a sequence of approximations to the true energy \(E\) and true wave function \(\psi\). The basic idea of perturbation theory is very simple: we split theHamiltonian into a piece we know how to solve (the ``reference'' or``unperturbed'' Hamiltonian) and a piece we don't knowhow to solve (the ``perturbation''). This result, that the first-order correction to the energy vanishes, could have been foreseen. [ "article:topic", "authorname:simonsj", "Perturbation Theory", "showtoc:no" ], Professor Emeritus and Henry Eyring Scientist (Chemistry), Telluride Schools on Theoretical Chemistry, information contact us at, status page at $\endgroup$ – JMJ May 4 '17 at 0:54 $\begingroup$ This is find ten eigenstate and corresponding values until 50episilon terms and to use pade-50/50 approximation to plot graph. Because the angular dependence of the perturbation (i.e., \(\cos \theta\)) has no \(\phi\)-dependence, matrix elements of the form, \[ \int ​Y_{J,M}^*(\theta,\phi) \cos\theta Y_{J,M}(\theta,\phi)\sin\theta d\theta d\phi=0\]. which is the RSPT expression for \(E^{(1)}\). In RSPT, one assumes that the only contribution of \(\psi^{(0)}\) to the full wave function \psioccurs in zeroth-order; this is referred to as assuming intermediate normalization of y. The effect of this perturbation on the energies is termed the Stark effect. 13.7). To solve a problem using perturbation theory, you start by solving the zero-order equation. First, one multiplies this equation on the left by the complex conjugate of the zeroth-order function for the state of interest \(\psi^{(0)}\) and integrates over the variables on which the wave functions depend. We introduce the expansion (4) as in Step A. In the HF model, one uses as a zeroth-order Hamiltonian, \[H^{(0)}=\sum_{i=1}^3 \left[\frac{1}{2}\nabla_i^2-\frac{3}{r_i}+V_{\rm HF}(r_i)\right]\], consisting of a sum of one-electron terms containing the kinetic energy, the Coulomb attraction to the nucleus (I use the Li atom as an example here), and a potential \(V_{\rm HF}(\textbf{r}_i)\). This means one needs to first form the 2x2 matrix, \[\left(\begin{array}{cc} k + Ç«. ϕ. The accuracy of the perturbation expansion is analyzed in detail by the discussion of an exactly solvable model. As long as the perburbation issmall compared to the unperturbed Hamiltonian, perturbation theorytells us how to correct the solutions to the unperturbed problem … Legal. We consider a one-dimensional quantum harmonic oscillator under the influence of a weak electric field of the form -eλx.. We use the Dirac formalism of Quantum Mechanics and write a,a † for the annihilation and creation operators of the harmonic oscillator. 4. Perturbation theory: Overview • We can use our understanding of vowel articulations as narrowings in the vocal tract… - to model expected deviations in the resonance frequencies from those of a uniform tube ([ə]) - and thereby predict formants of non-[ə] vowels • Later in the course, we will also use perturbation theory to This provides an approximate solution consisting of E 0 and ψ 0. … Hence, one writes the energy \(E\) and the wave function \(\psi\) as zeroth-, first-, second, etc, order pieces which form the unknowns in this method: \[E = E^{(0)} + E^{(1)} + E^{(2)} + E^{(3)} + \cdots\], \[y = \psi^{(0)} + \psi^{(1)} + \psi^{(2)} + \psi^{(3)} + \cdots\].

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