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. J

    Expected value of complex gaussian

    What is the expected value of the following expression exp(|z+\mu|), where \mu is a real constant and z=x+jy such that x and y are independent Gaussian random variables each with zero mean and \sigma^2 variance. When I try to take the expectation, I couldn't obtain a gaussian integral, so I...
  2. J

    Definite integral of gaussian

    Is there a closed form expression for the following definite integral? \int_{-∞}^{∞} exp(\frac{-|z|^2}{2{\sigma}^2}-\alpha |\mu + z|)dz where z is complex, and \alpha, \sigma, \mu are real constants. I couldn't obtain an expression similar to Gaussian integral, so I couldn't take the...
  3. E

    Continous Time Gaussian Distribution

    Hello all, I have the following equation \mathbf{v}(t)=\mathbf{P}(t)\mathbf{d}+\mathbf{w}(t) where v(t) is a 2-by-1 vector, P(t) is 2-by-2N matrix, d is a 2N-by-1 vector, and w(t) is an 2-by-1 Gaussian process vector where each element is of zero mean and variance N0. What is the probability...
  4. I

    Modifying a Gaussian Elimination Algorithm to Perform Gauss-Jordan E.

    Homework Statement I have an algorithm that implements Gaussian elimination. According to the text, with some modification of the indices and their in the loops, I should be able to have this algorithm perform Gauss-Jordan elimination. I also have to reduce the matrix to reduced row-echelon...
  5. M

    Exploring Holographic Hierarchy in the Gaussian Matrix Model

    This paper seems to be another entry in the LQG/string crossover competition: http://arxiv.org/abs/1303.3246 Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere Authors: David Garner, Sanjaye Ramgoolam (Submitted on 13 Mar 2013) The Gaussian Hermitian matrix model was...
  6. binbagsss

    Radioactive Decay - Gaussian or Poisson

    Radioactive Decay Probability? Say you are counting the number of decays and the time of observation is varied. I know that as time increases, the Gaussian Distribution becomes a closer fit to the observed probability than when the time interval takes smaller values because the mean count...
  7. B

    Top view of a Gaussian surface in a uniform electric field

    The field makes an angle θ with side 1 and the area of each face is A. In symbolic form, find the electric flux through (a) face 1, (b) face 2, (c) face 3, (d) face 4 and (e) top and bottom. My professor got: a=EAcosθ b=-EAsinθ c=-EAcosθ d=EAsinθ e= 0 I understant why e=0 but for the...
  8. skate_nerd

    Finding the Electric Field Outside of a Spherical Shell Using Gauss's Law

    Homework Statement The problem states that you have a spherical shell with inner radius Ri=1 cm and outer radius R0=2 cm. The shell also has uniform charge density of ρ=10-3 N/m3. I found the first few answers of the question already. First was to get the charge of the shell, which is simply...
  9. L

    Volume of a three dimensional gaussian

    Homework Statement How can I find the volume of a three dimensional gaussian exp\left [ \frac{-x^2}{\sigma_{x}} \frac{-y^2}{\sigma_{x}}\frac{-z^2}{\sigma_{z}} \right ] ? Since it is a gaussian, the volume should actually extend to infinity. It seems like there should be a simple double or...
  10. N

    How to interpret quotient rings of gaussian integers

    Homework Statement This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on. What are some of the elements of Z[i]/I where I is an ideal generated by a non-zero non-unit integer. For the sake of argument...
  11. 7

    Gaussian function to derive Heisenberg's uncertainty principle

    At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves ##\psi = \psi_0(k) e^{i(kx - \omega t)}## over wave numbers ##k##. We do it at ##t=0## hence ##\psi = \psi_0(k) e^{ikx}## \psi = \int\limits_{-\infty}^{+\infty} \psi_0\!(k) \cdot...
  12. P

    Gaussian Integral Substitution

    If I had an integral \int_{-1}^{1}e^{x}dx Then performing the substitution x=\frac{1}{t} would give me \int_{-1}^{1}-e^\frac{1}{t}t^{-2}dt Which can't be right because the number in the integral is always negative. Is this substitution not correct? Sorry if I am being very thick but I...
  13. G

    Calculating Flux Through a Gaussian Spherical Shell Inside a Charged Sphere

    Homework Statement An insulating sphere of radius R has positive charge uniformly distributed throughout its volume. The volume charge density (i.e., the charge per volume) is ρ. What is the flux through a Gaussian spherical shell of radius R/2 that is totally contained inside the charged...
  14. B

    Net electrical flux on a cylindrical Gaussian Surface

    Homework Statement a cylindrical solid of charge q, radius R, and length H. The Gaussian surface S is a cylindrical shell of radius r and length h, with r < R. Determine the net electric flux given that q = -48Q, R = 4L, H = 3L, r = 2L, and h = 2L (type the integer value, along with the sign...
  15. W

    Binary Detection in Gaussian Noise

    I have a vector signal, \underline{x}(t), which is afflicted with Gaussian noise \underline{n}(t). I take a finite number, L, of discrete observations and (based on those observations) want to determine whether: (1) Only Gaussian noise is present, \left[\text{i.e. } \underline{x}(t) =...
  16. C

    What is the total flux flowing through a spherical Gaussian surface?

    Homework Statement Consider a uniformly charged sphere (an insulating sphere of radius R,) and a spherical Gaussian surface with radius R/2 concentric to the sphere. What is the total flux flowing through the Gaussian surface? Homework Equations Vsphere= (4∏R^3)/3 Asphere= 4∏R^2 Gauss' Law...
  17. R

    What are the two sides that contribute to the divergence theorem?

    Homework Statement Suppose the one-dimensional field A = Kx * ax exists in a region. Illustrate the validity of the Gaussian theorem by evaluating its volume and surface integrals inside and on the rectangular parallelepiped bounded by the surfaces: x=1,x=4,y=2,y=-2,z=0 and z=3, for a given...
  18. C

    Fourier Transform of a Gaussian Pulse

    Homework Statement Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum. Homework Equations The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by f(t) = exp(-t^2/2)...
  19. F

    Average of Dirac Delta-Function over Double Gaussian Variables

    I need to work out an expression for the average of a Dirac delta-function \delta(y-y_n) over two normally distributed variables: z_m^{(n)}, v_m^{(n)} So I take the Fourier integral representation of the delta function: \delta(y-y_n)=\int \frac{d\omega}{2\pi} e^{i\omega(y-y_n)} =\int...
  20. Q

    (Quantum Mechanics) Gaussian Distributions, Expected Values, and Sketches

    Homework Statement Consider the gaussian distribution ρ(x) = Aexp[(-λ^2)(x-a)^2] , where A, a, and λ are positive real constants. (a) Find A such that the gaussian distribution function is normalized to 1. (b) Find <x> (average; expected value) , <x^2>, and σ (standard deviation). (c)...
  21. C

    How Does Moving a Charge Affect Electric Flux in a Cubic Gaussian Surface?

    Homework Statement A point charge of negative polarity is located at the centre of a cubic Gaussian surface with edges of length ##0.5m##. Calculate the electric flux through one of the faces of the surface. What would happen if the charge was moved 10cm to the right? Homework Equations...
  22. S

    What is the electric field in a thin slab with uniform charge distribution?

    Homework Statement 4.10.4 Thin Slab Let some charge be uniformly distributed throughout the volume of a large planar slab of plastic of thickness d. The charge density is ρ. The mid-plane of the slab is the y-z plane. (a) What is the electric field at a distance from the mid-plane when |x| <...
  23. S

    Gaussian vs SI Units: Comparison & Advantages

    Hello, I was reading about a comparison betw. Gaussian and SI units and the author states "The main advantage of Gaussian units is that they make fundamental physical issues and theoretical relations involving electromagnetic phenomena more clear." Would someone know - is the advantage...
  24. K

    Gaussian integrals with 4-momentum

    1. Gaussian Please help me calculate some Gaussian integrals in the attached file which are used in my QFT calculations. Thank you very much!
  25. K

    Gaussian integrals with 4-momentum

    I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!
  26. A

    Using a cube as a Gaussian surface

    Homework Statement Suppose there is a uniformly charged cube with known side length. I then imagine a larger, closed cube surface surrounding it. This larger cube has double the side length and is symmetrical to the smaller cube. Is is practical to use this Gaussian surface to compute the...
  27. A

    Solving Gaussian Potential - Analyzing Energy Differences

    Okay, I have solved the schrödinger equation numerically by making it dimensionless (though I am still confused about this proces). And then approximating it on a finite interval and solving the resulting eigenvalue equation. This allows me to solve for the wave function of different potentials...
  28. D

    Frequency doubling and gaussian beam problem.

    1. A Nd:YAG laser operates at a wavelength of 1064 nm and is used for frequency doubling in the non-linear material LiNbO3. The laser outputs 1 W of power in a TEM00 mode with a beam waist ω0 = 1.0 mm on the output mirror. A lens of focal length of 50 mm is placed immediately outside the...
  29. Mandelbroth

    Derivation of the Antiderivative of the Gaussian Distribution

    I'm in a high school pre-calculus class and a statistics class. For the latter, we are given z-tables for some of our tests. I don't like these z-tables. Thus, I decided that a more direct approach (fundamental theorem of calculus) would be more accurate and, more importantly, more fun. My...
  30. G

    Change of variable in integral of product of exponential and gaussian functions

    I have the integral \int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2} where \xi is a constant. I would like to transform by some change of variables in the form \int_{-\infty}^{\infty}dx F(x) \int_{-\infty}^{\infty}dy G(y) the problem is...
  31. jk22

    Exploring Dirac Diffusion: Gaussian Solutions?

    does anyone know how dirac diffusion looks like, i.e. The solution of dirac equation with no potential and initial condition for 1 spinor component psi(x,t0) being delta(x-x0) ? Are the solution gaussian like the schroedinger case ? Thanks.
  32. A

    Finding the Normalization Constant of a Gaussian Distribution (Griffiths 1.6)

    Homework Statement Consider the Gaussian Distribution ρ(x) = A e^{-λ(x-a)^{2}} where A, a, and λ are constants. Determine the normalization constant A. Homework Equations \int^{∞}_{-∞}ρ(x) dx = 1 The Attempt at a Solution The problem recommends you look up all necessary integrals, so I...
  33. J

    Intuitive explanations for Gaussian distribution function and mahalanobis distance

    Hello I was wondering If anyone could give intuitive explanations for the multivariate Gaussian distribution function and mahalanobis distance? My professor didn't explain these in probability class, they were only defined... Where did the formula come from? Why is the Gaussian function the...
  34. P

    Rearranging a formula (Multivariate Gaussian function)

    Homework Statement See image, p(y|θ) is the Likelihood function which has to be rearranged in the form of equation (3). θ is a vector variable. Homework Equations None? The Attempt at a Solution I first expanded the exponent in the original function, equation (2)...
  35. M

    Modeling Public Bus Headways with Gaussian Processes

    Please help me find some references where public bus headways are described(modeled) as gaussian processes. Best Regards
  36. dexterdev

    How to convert a gaussian distributed rnd sequence to uniform dist and viceversa

    Hi PF, I have a device (with a microcontroller) which generates random numbers. when I analyzed those numbers using Matlab software I found that it is following a uniform distribution. How can I mathematically (any algorithm?) convert this random output to a gaussian one. Also I would...
  37. P

    Similar problem to Gaussian integral

    We all know about the famous equation: \int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}. How about \int_{-\infty}^\infty e^{-x^4} dx? Or, in general, can we calculate any integral in the form \int_{-\infty}^\infty e^{-x^n} dx, where n is an even counting number?
  38. L

    Gaussian Radius of curvature in an argon ion laser

    Hi, having a bit of trouble with this question "In an argon ion laser ( λ = 514nm) the minimum beam waist is 1.0mm and is close to the plane mirror. Calculate the radius of curvature of the beam at the output mirror. 1.15m away" Attempt at a solution: θ = 2λ/pi W02 R = Z + ZR2/Z ZR =...
  39. A

    Is it possible to use complex contour integrals to solve gaussian integrals?

    I have an integral of the form (from -inf to inf): ∫exp((a+ib)x2+icx)dx How do I solve that? I have tried setting y = √(a+ib)(x+ic/√(a+ib)) And you then get an integral of exp(-y2) times a constant. But I don't even know if this is legal since part of the exponentials are complex...
  40. M

    Fourier Transform of a Gaussian With Non-Zero Mean

    Homework Statement I am looking at finding the Fourier transform of: f(t)=\exp \left[ \frac{-(t-m)^2}{2 \sigma^2}\right] Homework Equations \hat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt The Attempt at a Solution I did it a little differently that my...
  41. M

    Equation for half-max contour of 2D Gaussian?

    Hi all, If I have a Gaussian with the equation: G(x,y) = h*exp(-(x-x0)^2/a -(y-y0)^2/b) where x0, y0, a, b and h are the parameters which may vary, what's the equation for the elliptical contour line at the half-max of G? I'm getting myself confused! Thanks for help
  42. G

    Solve Problems with Gaussian Random Matrix: Compressed Sensing Senior Project

    I'm doing my senior project about Compressed sensing, I have some problems with some algorithms. 1)Can anybody help me about Gaussian random matrix, how we can explain it briefly. 2)Does the randn(m,n) building function in Matlab working bases on Probability density function? How does randn...
  43. M

    Need help with Orca, gaussian, gamess.

    hey can anyone give me some literature on how work with those programs (Orca, gaussian, gamess) ? I need to do optimization on large quantum system with those programs but at my university i can not find any literature.
  44. M

    Where can i learn how Gaussian calculations various molecular properties?

    I have a book on Gaussian but it mainly just explains how to use the program, it only gives very brief explanations of the theory behind how Gaussian does what it does. I want to gain an understanding of how Gaussian performs the various types of calculations that it can do. I understand the...
  45. M

    Does the Gaussian have complex poles?

    I know that \int_{- \infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}. What if I tried evaluating this over the complex plane? If I created a contour that ran along the real axis, then circled around from positive to negative infinity this should not change things as along the curve at...
  46. M

    Simulating Gaussian Process in R w/ Mean & Variance

    Hi everybody Any ideas how to simulate a Gaussian stationary process in R language using predefined variance and mean? I have a uni-variate normal distribution for my real life process Thank you
  47. dexterdev

    Why do we consider zero mean for gaussian noise in a communication channel?

    Hi all, I got an idea why noise is having a gaussian pdf but didnot understood why it should be having zero mean. Why can't noise contain a dc content. -Devanand T
  48. M

    Gaussian - Using Hybrid Functionals

    I read that B3LYP is a hybrid functional which uses some HF method and some DFT method for its calculations. According to this page: my professor told me that you can set the proportions of each method that the B3LYP uses yourself, so for example you can make it so it uses 70% HF and 30% DFT or...
  49. T

    Derivation of Gaussian formula forom particular situation

    Homework Statement We are to derive the Gaussian formula for a case with a virtual object, a virtual image, and a concave surface. Homework Equations Snell's law, n(sin(phi)) = n'(sin(phi')) geometry The Attempt at a Solution I can do a derivation for a situation with a...
  50. fluidistic

    Marginal distribution of a Gaussian

    Homework Statement The random variables X and Y have a joint probability distribution of f_{XY}(x,y)= \frac{1}{2\pi \sigma _X \sigma _Y \sqrt{1-\rho ^2}} \exp \{ \left [ -\frac{1}{2(1-\rho ^2)} \right ] \left [ \left ( \frac{x-\mu _X}{\sigma _X} \right )^2 + \left ( \frac{y-\mu...
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