What is Gaussian: Definition and 771 Discussions

Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70. It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16. Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.

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  1. M

    I Gaussian beam in a Fabry-Perot interferometer

    Hello I am reading some introductory laser cavity stuff and I am a bit confused about the existence of gaussian beams in the Fabry-Perot interferometer. If you solve the stability condition for a cavity (i.e. asking for the q parameter to reproduce itself after one round trip) you get that in...
  2. L

    MATLAB Gaussian process and climate model in Matlab

    I'll admit I am very new to Gaussian processes, but from what I know a Gaussian process is completely determined by a mean vector E(Y(θ)) and a covariance function Cov[Y(θ1), Y(θ2)]. E(Y(θ)) is given, and we have the correlation, which is just the covariance divided by Var(θ1)*Var(θ2).The...
  3. Shackleford

    Is My 2D Gaussian Quadrature Algorithm Accurate?

    ## \int_{-1}^{1} \int_{-1}^{1} e^{-(x^2 + y^2)} cos(2π (x^2 + y^2)\,dx\,dy ## ## I = \int_{-1}^{1} \int_{-1}^{1}f(x,y) \,dx\,dy \approx \sum_{i=0}^{n}\sum_{j=0}^{n} w_i w_j f(x_i, y_j) ## ## = w_0 w_0 f(x_0, y_0) + w_0 w_1 f(x_0, y_1) + w_1 w_0 f(x_1, y_0) + w_1 w_1 f(x_1, y_1) ## ## w_0 =...
  4. E

    Can Math Model Real-World Camera Focusing Dynamics?

    This problem arose in modeling camera focusing movement, such as a control system might do. It assumes a simple (thin) lens, rays close to the optical axis, and monochromatic light. While most camera lenses are not simple, this is a first approximation. Camera lenses project an image of a...
  5. cianfa72

    I Gaussian elimination for homogeneous linear systems

    Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...
  6. H

    I Does a Gaussian wave packet remain Gaussian?

    Consider a gaussian wave packet whose wave function at a particular instant of time is Its time dependence is implicit in the "constants" A, a, <x> and <p>, which may all be functions of time. But regardless of what functions of time they may be, these constants will take on some values at...
  7. cianfa72

    I Gaussian elimination for a singular square matrix

    Hi, I've the following doubt: consider an homogeneous linear system ##Ax=0## with ##A## a singular square matrix. The resulting matrix attained through Gaussian elimination will be in upper triangular or raw echelon form ? Thanks.
  8. M

    A Product of Gaussian and Rayleigh distributions gives what distribution?

    Hello, I'm trying to find out the distribution function (cumulative or density) of the product of two independent random variables respectively following a non-zero-mean Gaussian and a Rayleigh distribution. The math is too intricate for me, I've found in the appendix of [Probability...
  9. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    Given two variables ##x## and ##k##, the covariance between the variables is as follows, where ##E## denotes the expected value: \begin{equation} \begin{split} COV(x,k)&= E[x k]-E[x]E[k] \end{split} \end{equation} If ##x## and ##k## are Foureir conjugates and ##f(x)## and ##\hat{f}(k)## are...
  10. koulbichok

    I Gaussian probability distribution of formation PBH

    Hello. If we consider PBH formation from collapse of large density perturbation in the early Universe, a mass PBH depends on density contrast as And δ must be larger then . Also we have β — an abundance of black holes, it's the ratio of the PBH energy density to the total energy density, this...
  11. looseleaf

    I Gaussian Integral Coordinate Change

    Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...
  12. J

    What is the maximum or Nyquist frequency of a Gaussian signal?

    Hello. I'm studying Fourier analysis. If we look at attached graph where Gaussian functions are transformed by Fourier analysis, we can find Gaussian functions in frequency domain have maximum value at 0 hertz. So I confused what is the Nyquist frequency at Gaussian signal. I need to know...
  13. M

    MHB Gaussian Quadrature Formula for Integrating Polynomials of Degree 6

    Hey! :o I want to calculate the integral $$\int_0^1\frac{1}{x+3}\, dx$$ with the Gaussian quadrature formula that integrates exactly all polynomials of degree $6$. The gaussian quadrature integrates exactly polynomials $\Phi (x)$ with maximum degree $2n-1$. In this case we consider $n=4$...
  14. M

    MHB Gaussian Quadrature: isolated roots

    In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function. Then there is the following question: Explain why isolated roots are allowed in the weight function. What exacly is meant by that? Could you explain that to me? What are...
  15. N

    I Determine P(t = x) if % of sample drawn from gaussian > t

    Sorry for the bad title, limited space a sample group of size n, as well as a number t,is drawn randomly from a normal distribution, if we have the number of people in the sample group bigger than t, can we determine a PDF function of what value t is? Are they any simplifications we can use to...
  16. A

    I Discrete to continuum Gaussian function

    I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...
  17. Boltzman Oscillation

    What is the integral of this Gaussian distribution?

    Homework Statement Find A in p(x) = Aexp(-λ(x-a)^2) by using the equation 1 = ∫ p(x)dxHomework Equations 1 = ∫p(x)dx The Attempt at a Solution I expand the power of the exponential and then extract the constant exponential to get: Aexp(λa^2) ∫exp(-λx^2)exp(2aλx)dx I don't know how to...
  18. S

    Casimir effect with Gaussian regulator

    Homework Statement Calculate the Casimir force in 1D using a Gaussian regulator. Homework EquationsThe Attempt at a Solution I reached a point where I need to evaluate a sum of the form $$\sum_n n e^{-\epsilon^2n^2}$$ Can someone help me? I didn't really find anything useful online. I thought...
  19. N

    A What is the variance of a Gaussian RV

    Hi, Let y = x + z, where x and z are mutually independent RVs. Also, z is a complex gaussian RV with zero mean and variance sigma^2. My question is as follows: For x = y - z, what is the variance of (-z) ? Any help could be useful. Thanks in advance.
  20. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...
  21. D

    I Finding expansion coefficient of a 3-d Gaussian wave packet

    I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet. First I'm decomposing a Gaussian wave packet $$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0...
  22. R

    Intensity of a Gaussian laser beam

    I have two lasers with different intensity distributions (shown below) — one is Gaussian and the other one is rectangular (having the shape of a Fresnel diffraction pattern at the target). I am trying to compare the efficacy of the two lasers for burning a certain material (I am really...
  23. Ken Gallock

    I Gaussian normal coordinates and Riemann normal coordinates

    Hi. I was wondering what is the relationship between Gaussian normal coordinates and Riemann normal coordinates. Thanks.
  24. corpsinhere

    I F(x, y) describing a distorted 3D Gaussian bell

    I am new to these forums - if I have posted in the wrong place please let me know. Standard 3D Gaussian bell: z = e^-(x^2) * e^-(y^2) From along the z-axis this looks "round". I would like a generalized f(x, y) which would look egg-shaped from above - possibly quite distorted.. I thought at...
  25. W

    I What is a Gaussian Wave Packet?

    Can anyone tell me what a Gaussian Wave Packet is? What happens to the atoms inside a Gaussian Wave Packet? Can more than one Gaussian Wave Packet Exist in the same place? Thank you,
  26. Ventrella

    A Differences between Gaussian integers with norm 25

    I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...
  27. G

    Area under the Gaussian Curve

    I have been teaching undergrad students informally, and one of the math problems that I have always enjoyed introducing them to is how to compute the area under a gaussian curve, or to keep it simple, the area under the curve ##z=e^{-x^2}## One of my students asked me a question that has...
  28. B

    Gaussian Smoothing of the Matter Power Spectrum

    Homework Statement Consider the density perturbation smoothed with a Gaussian of scale ##\sigma##, ##\Delta_{\sigma}(\vec x') = \int d^3 \vec x \frac{e^{- \frac{(\vec x - \vec x')^2}{2 \sigma^2}}}{(2 \pi \sigma)^{3/2}} \Delta (\vec x)## Calculate the power spectrum ##P_{\Delta_{\sigma}}## of...
  29. A

    I Sampling from a multivariate Gaussian distribution

    I was watching a lecture on youtube about linear regression and there's a section where it had the statement below (written in purple). Does multiplying by sigma rotate the distribution to make it look like x - N(mew, sigma^2)? Mew in this case is 0 so it doesn't shift the distribution.
  30. NoahCygnus

    Electric field on Gaussian surface due to external charge

    There's something I need to confirm about Gauss' law. If I have to determine the electric field at point P due to charge +q, I take a Gaussian sphere enclosing the charge with the point on the surface of the sphere. So Gauss law doesn't care about the charge +Q because the flux do to this charge...
  31. ohwilleke

    I Why do particle physicists use Gaussian error estimates?

    There is solid empirical evidence that error in particle physics measurements is not actually distributed in a Guassian manner. Why don't particle physicists routinely use student t error distributions with fat tails that fit the reality of errors in experimental measurement more accurately...
  32. R

    B Exponential decay convolved with Gaussian

    Hello all, I have a data which look like reversed exponentially modified Gaussian (EMG) function and interested to fit the data with with reversed EMG function. After searching on internet I found the EMG function, which is given below...
  33. D

    I On the Gaussian Curvature of time-like surfaces

    Firstly, I am asking for your patience and understanding because my maths formalism is not going to be rigorous. In another thread here in this forum, I set an example for which now I am asking further instructions. I am going to ask about time-like surfaces immersed in Minkowskian space-time...
  34. M

    Advantages of Hermite-Gaussian beams?

    And areas of usage? I will be glad if you help me.
  35. C

    I Integrate translated gaussian function by change of variable

    I want to integrate this: \int_0^∞ re^{-\frac{1}{2σ^2} (r-iσ^2q)^2} \, dr. If I change the variable r into t with this relation:r-iσ^2q=t, then the integral becomes\int_{-iσ^2q}^∞ (t+iσ^2q)e^{-\frac{1}{2σ^2} t^2} \, dt so it seems I cannot use the famous gaussian integral formula. But I got the...
  36. T

    Show the Fourier transformation of a Gaussian is a Gaussian.

    Homework Statement Show, by completing the square in the exponent, that the Fourier transform of a Gaussian wavepacket ##a(t)## of width ##\tau## and centre (angular) frequency ##\omega_0##: ##a(t)=a_0e^{-i\omega_0t}e^{-(t/\tau)^2}## is a Gaussian of width ##2/\tau##, centred on ##\omega_0##...
  37. tarkin

    QM harmonic oscillator - integrating over a gaussian?

    Homework Statement [/B] For the first excited state of a Q.H.O., what is the probability of finding the particle in -0.2 < x < 0.2 Homework Equations Wavefunction for first excited state: Ψ= (√2) y e-y2/2 where: The Attempt at a Solution To find the probability, I tried the integral of...
  38. O

    I Gaussian Quadrature on a Repeated Integral

    Hi there, I am having some difficulty evaluating a repeated integral, which is the first of two shown in the image. I had hoped to be able to use Gaussian Quadrature to provide a numerical result, however am unsure on if this is possible for a repeated integral? I have attempted to use Cauchy'...
  39. O

    Gaussian type integral (but not a standard form)

    When working a proof, I reached an expression similar to this: $$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$ I've tried the following: 1. I tried squaring and combining and converting to polar coordinates, like one would solve a standard Gaussian. However...
  40. M

    Coefficients that make Gaussian elimination impossible?

    Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...
  41. W

    QM: Difference between these Initial Wavefunctions

    Homework Statement I've been asked as a part of some school project to find the Fourier transform, and time evolution of the following initial wavefunctions: 1. ##\Psi(x,0) = Ae^{\frac{-x^2}{2\sigma ^2}}## 2. ##\Psi(x,0) = Be^{\frac{-x^2}{2\sigma ^2}}e^{\frac{ipx}{\hbar}}## What physical...
  42. M

    Why does Gaussian elimination work?

    Post moved by moderator from Homework section Hello, I was curious as to why Gaussian elimination works. I know that if we have two ( or more) systems of two (or more) linear equations, we can write then in terms of a matrix. However, what does it mean when I get the identity on the left hand...
  43. D

    Electric Flux through Cubical Surface Enclosing Sphere

    Homework Statement A uniform charge density of 700 nC/m3 is distributed throughout a spherical volume of radius 6.00 cm. Consider a cubical Gaussian surface with its center at the center of the sphere. [reference picture] What is the electric flux through this cubical surface if its edge...
  44. B

    Gaussian beam spherical mirror reflection question

    Homework Statement Gaussian beam of radius R_i and beam width w_i, The beam is reflected off a mirror with a radius of curvature R = R_i and the reflectivity of this mirror is given as rho(r) = rho_0*exp(-r^2/a^2), where r is the radial distance from the center of the mirror and a is a...
  45. A

    I Negative values for Gaussian Distribution

    So in my Physics lab, we divided into groups and our task was to throw darts on a target containing 13 bins. The bins look something like the image below. At the end, our class combined our average, standard deviation, and standard error. I made a Gaussian Distribution and I noticed that the...
  46. N

    Cross-correlation of white noise process with its conjugate

    If w[n] are samples of the white gaussian noise process, I know that E[w[n1] w[n2]] = 0 for a WGN process. what would the following expression lead to: E[w[n1] w*[n2]] = ? Would it also be zero? Thanks a lot!
  47. A

    Mean squared value of the Gaussian

    Homework Statement If ##P(x)\propto e^{x^2/2\sigma^2}##, show that the average ##\langle x^2\rangle = \sigma^2##. Homework Equations ##\langle x\rangle = \frac {\int xP(x) \, dx} {\int P(x) \, dx}## The Attempt at a Solution ##I = \int x^2e^{x^2/2\sigma^2} \, dx = \int (-\sigma^2x)(\frac...
  48. Isaac0427

    B Uncertainty of a Gaussian wavepacket

    Hi, I know that a Gaussian wavepacket has minimum uncertainty. The issue is, some sources are telling me that σxσp=ħ and others are telling me that σxσp=ħ/2. I am really confused. I think the latter is correct due to what I have been taught about the uncertainty principle, but then I don't...
  49. Sunbodi

    When to Integrate Charge Enclosed for Gaussian Surfaces?

    Hello, I was looking over my notes and I was trying to figure out when we integrate Q enclosed when Q = ρ*d(volume). If there's one thing I've learned from physics II you only integrate when a field is non-uniform. I'm just wondering how we know when it's uniform (usually the problem will tell...
  50. C

    MHB Are components in a gaussian mixture independent?

    Hello all, I have used Expectation Maximization algorithm to approximate a probability density function (pdf) using a mixture of gaussians. I need to square the pdf corresponding to the mixture of gaussians( it is a weighted sum of normal distributions with non-identical parameters). Rather...
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